Chapter 1
Cartesian Coordinate Systems
Before turning to those moral and mental aspects of the matter which
present the greatest difficulties, let the inquirer begin by mastering
more elementary problems.
— Sherlock Holmes from A Study in Scarlett (1887)
3D math is all about measuring locations, distances, and angles precisely and mathematically in 3D
space. The most frequently used framework to perform such calculations using a computer is called
the Cartesian coordinate system. Cartesian mathematics was invented by (and is named after)
a brilliant French philosopher, physicist, physiologist, and mathematician named
René Descartes, who lived from 1596 to 1650. René Descartes is famous not just for
inventing Cartesian mathematics, which at the time was a stunning unification of algebra and
geometry. He is also wellknown for making a pretty good stab of answering the question
“How do I know something is true?”—a question that has kept generations of philosophers happily
employed and does not necessarily involve dead sheep (which will perhaps disturbingly be a central
feature of the next section), unless you really want it to. Descartes rejected the answers proposed
by the Ancient Greeks, which are ethos (roughly, “because I told you so”), pathos
(“because it would be nice”), and logos (“because it makes sense”), and set about
figuring it out for himself with a pencil and paper.
This chapter is divided into four main sections.

Section 1.1 reviews some basic principles of number systems and the first law of
computer graphics.

Section 1.2 introduces 2D Cartesian mathematics, the mathematics of flat
surfaces. It shows how to describe a 2D cartesian coordinate space and how to locate
points using that space.

Section 1.3 extends these ideas into three dimensions. It explains
left and righthanded coordinate spaces and establishes some conventions used in this
book.

Section 1.4 concludes the chapter by quickly reviewing assorted
prerequisites.
1.11D Mathematics
You're reading this book because you want to know about 3D mathematics, so you're probably
wondering why we're bothering to talk about 1D math. Well, there are a couple of issues about
number systems and counting that we would like to clear up before we get to 3D.
Figure 1.1One dead sheep
The
natural numbers, often called the counting numbers, were invented millennia ago,
probably to keep track of dead sheep. The concept of “one sheep” came easily (see
Figure 1.1), then “two sheep,” “three sheep,” but people very quickly became convinced
that this was too much work, and gave up counting at some point that they invariably called “many
sheep.” Different cultures gave up at different points, depending on their threshold of boredom.
Eventually, civilization expanded to the point where we could afford to have people sitting around
thinking about numbers instead of doing more survivaloriented tasks such as killing sheep and
eating them. These savvy thinkers immortalized the concept of zero (no sheep), and although they
didn't get around to naming all of the natural numbers, they figured out various systems whereby
they could name them if they really wanted to using digits such as 1, 2, etc. (or if you were
Roman, M, X, I, etc.). Thus, mathematics was born.
The habit of lining sheep up in a row so that they can be easily counted leads to the concept of a
number line, that is, a line with the numbers marked off at regular intervals, as in
Figure 1.2. This line can in principle go on for as long as we wish, but to avoid boredom
we have stopped at five sheep and used an arrowhead to let you know that the line can continue.
Clearer thinkers can visualize it going off to infinity, but historical purveyors of dead sheep
probably gave this concept little thought outside of their dreams and fevered imaginings.
At some point in history, it was probably realized that sometimes, particularly fast talkers could
sell sheep that they didn't actually own, thus simultaneously inventing the important concepts of
debt and negative numbers. Having sold this putative sheep, the fast talker would in fact own
“negative one” sheep, leading to the discovery of the
integers, which consist of the natural numbers and their negative counterparts. The
corresponding number line for integers is shown in Figure 1.3.
The concept of poverty probably predated that of debt, leading to a growing number of people who
could afford to purchase only half a dead sheep, or perhaps only a quarter. This led to a
burgeoning use of fractional numbers consisting of one integer divided by another, such as 2/3 or
111/27. Mathematicians called these
rational numbers, and they fit in the number line in the obvious places between the
integers. At some point, people became lazy and invented decimal notation, writing “3.1415”
instead of the longer and more tedious 31415/10000, for example.
After a while it was noticed that some numbers that appear to turn up in everyday life were not
expressible as rational numbers. The classic example is the ratio of the circumference of a circle
to its diameter, usually denoted
$\pi $
(the Greek letter pi, pronounced “pie”). These are the
socalled
real numbers, which include the rational numbers and numbers such as
$\pi $
that would, if
expressed in decimal notation, require an infinite number of decimal places. The mathematics of
real numbers is regarded by many to be the most important area of mathematics—indeed, it is the
basis of most forms of engineering, so it can be credited with creating much of modern
civilization. The cool thing about real numbers is that although rational numbers are countable
(that is, can be placed into onetoone correspondence with the natural numbers), the real
numbers are uncountable.
The study of natural numbers and integers is called discrete mathematics, and the study of
real numbers is called continuous mathematics.
The truth is, however, that real numbers are nothing more than a polite fiction. They are a
relatively harmless delusion, as any reputable physicist will tell you. The universe seems to be
not only discrete, but also finite. If there are a finite amount of discrete things in the
universe, as currently appears to be the case, then it follows that we can only count to a certain
fixed number, and thereafter we run out of things to count on—not only do we run out of dead
sheep, but toasters, mechanics, and telephone sanitizers, too. It follows that we can describe the
universe using only discrete mathematics, and only requiring the use of a finite subset of the
natural numbers at that (large, yes, but finite). Somewhere, someplace there may be an alien
civilization with a level of technology exceeding ours who have never heard of continuous
mathematics, the fundamental theorem of calculus, or even the concept of infinity; even if we
persist, they will firmly but politely insist on having no truck with
$\pi $
, being perfectly happy to
build toasters, bridges, skyscrapers, mass transit, and starships using 3.14159 (or perhaps
3.1415926535897932384626433832795 if they are fastidious) instead.
So why do we use continuous mathematics? Because it is a useful tool that lets us do engineering.
But the real world is, despite the cognitive dissonance involved in using the term “real,”
discrete. How does that affect you, the designer of a 3D computergenerated virtual reality? The
computer is, by its very nature, discrete and finite, and you are more likely to run into the
consequences of the discreteness and finiteness during its creation than you are likely to in the
real world. C++ gives you a variety of different forms of number that you can use for counting or
measuring in your virtual world. These are the
short, the
int, the
float and the
double, which can be described as follows (assuming current PC technology). The short
is a 16bit integer that can store 65,536 different values, which means that “many sheep” for a
16bit computer is 65,537. This sounds like a lot of sheep, but it isn't adequate for measuring
distances inside any reasonable kind of virtual reality that take people more than a few minutes to
explore. The int is a 32bit integer that can store up to 4,294,967,296 different
values, which is probably enough for your purposes. The float is a 32bit value that
can store a subset of the rationals (slightly fewer than 4,294,967,296 of them, the details not
being important here). The double is similar, using 64 bits instead of 32.
The bottom line in choosing to count and measure in your virtual world using ints,
floats, or doubles is not, as some misguided people would have it, a
matter of choosing between discrete shorts and ints versus continuous
floats and doubles; it is more a matter of precision. They are all
discrete in the end. Older books on computer graphics will advise you to use integers because
floatingpoint hardware is slower than integer hardware, but this is no longer the case. In fact,
the introduction of dedicated floating point vector processors has made floatingpoint arithmetic
faster than integer in many common cases. So which should you choose? At this point, it is probably
best to introduce you to the first law of computer graphics and leave you to think about it.
The First Law of Computer Graphics
If it
looks right, it
is right.
We will be doing a lot of trigonometry in this book. Trigonometry
involves real numbers such as
$\pi $
, and realvalued functions such
as sine and cosine (which we'll get to later). Real numbers are a
convenient fiction, so we will continue to use them. How do you know
this is true? Because, Descartes notwithstanding, we told you so,
because it would be nice, and because it makes sense.
1.22D Cartesian Space
You probably have used 2D Cartesian coordinate systems even if you have never heard the term
“Cartesian” before. “Cartesian” is mostly just a fancy word for “rectangular.” If you have
ever looked at the floor plans of a house, used a street map, seen a football game, or played chess, you have some exposure to 2D Cartesian coordinate spaces.
This section introduces 2D Cartesian mathematics, the mathematics
of flat surfaces. It is divided into three main subsections.

Section 1.2.1 provides a gentle
introduction to the concept of 2D Cartesian space by imagining a
fictional city called Cartesia.

Section 1.2.2 generalizes this concept to arbitrary or abstract 2D
Cartesian spaces. The main concepts introduced are

the origin

the
$x$
 and
$y$
axes

orienting the axes in 2D

Section 1.2.3 describes how to
specify the location of a point in the 2D plane using Cartesian
$(x,y)$
coordinates.
1.2.1An Example: The Hypothetical City of Cartesia
Let's imagine a fictional city named Cartesia. When the Cartesia city planners were laying out the
streets, they were very particular, as illustrated in the map of Cartesia in
Figure 1.4.
As you can see from the map, Center Street runs eastwest through the middle of town. All other
eastwest streets (parallel to Center Street) are named based on whether they are north or south of
Center Street, and how far they are from Center Street. Examples of streets that run eastwest are
North 3rd Street and South 15th Street.
The other streets in Cartesia run northsouth. Division Street runs northsouth through the middle
of town. All other northsouth streets (parallel to Division Street) are named based on whether
they are east or west of Division Street, and how far they are from Division Street. So we have
streets such as East 5th Street and West 22nd Street.
The naming convention used by the city planners of Cartesia may not be creative, but it certainly
is practical. Even without looking at the map, it is easy to find the donut shop at North 4th and
West 2nd. It's also easy to determine how far you will have to drive when traveling from one place
to another. For example, to go from that donut shop at North 4th and West 2nd, to the police
station at South 3rd and Division, you would travel seven blocks south and two blocks east.
1.2.2Arbitrary 2D Coordinate Spaces
Before Cartesia was built, there was nothing but a large flat area of land. The city planners
arbitrarily decided where the center of town would be, which direction to make the roads run, how
far apart to space the roads, and so forth. Much like the Cartesia city planners laid down the
city streets, we can establish a 2D Cartesian coordinate system anywhere we want—on a piece of
paper, a chessboard, a chalkboard, a slab of concrete, or a football field.
Figure 1.5 shows a diagram of a 2D Cartesian coordinate system.
As illustrated in Figure 1.5, a 2D Cartesian coordinate
space is defined by two pieces of information:

Every 2D Cartesian coordinate space has a special
location, called the origin, which is the “center” of the
coordinate system. The origin is analogous to the center of
the city in Cartesia.

Every 2D Cartesian coordinate space has two straight lines
that pass through the origin. Each line is known as an
axis and extends infinitely in two opposite directions. The two
axes are perpendicular to each other. (Actually, they don't
have to be, but most of the coordinate systems we will look at
will have perpendicular axes.) The two axes are analogous to
Center and Division streets in Cartesia. The grid lines in the
diagram are analogous to the other streets in Cartesia.
At this point it is important to highlight a few significant differences between Cartesia and an
abstract mathematical 2D space:

The city of Cartesia has official city limits. Land outside of the city limits is
not considered part of Cartesia. A 2D coordinate space, however, extends infinitely.
Even though we usually concern ourselves with only a small area within the plane
defined by the coordinate space, in theory this plane is boundless. Also, the roads in
Cartesia go only a certain distance (perhaps to the city limits) and then they stop. In
contrast, our axes and grid lines extend potentially infinitely in two directions.

In Cartesia, the roads have thickness. In contrast, lines in an abstract
coordinate space have location and (possibly infinite) length, but no real thickness.

In Cartesia, you can drive only on the roads. In an abstract coordinate space,
every point in the plane of the coordinate space is part of the coordinate space, not
just the “roads.” The grid lines are drawn only for reference.
In Figure 1.5, the horizontal axis is called the
$x$
axis, with positive
$x$
pointing to the right, and the vertical axis is the
$y$
axis, with positive
$y$
pointing up. This
is the customary orientation for the axes in a diagram. Note that “horizontal” and
“vertical” are terms that are inappropriate for many 2D spaces that arise in practice. For
example, imagine the coordinate space on top of a desk. Both axes are “horizontal,” and
neither axis is really “vertical.”
The city planners of Cartesia could have made Center Street run northsouth instead of eastwest.
Or they could have oriented it at a completely arbitrary angle. For example, Long Island, New York,
is reminiscent of Cartesia, where for convenience the “streets” (1st Street, 2nd Street etc.) run
across the island, and the “avenues” (1st Avenue, 2nd Avenue, etc.) run along its long axis. The
geographic orientation of the long axis of the island is an arbitrary result of nature. In the
same way, we are free to orient our axes in any way that is convenient to us. We must also decide
for each axis which direction we consider to be positive.
For example, when working with images on a computer screen, it is customary to use the coordinate
system shown in Figure 1.6. Notice that the origin is in the upper lefthand
corner,
$+x$
points to the right, and
$+y$
points down rather than up.
Unfortunately, when Cartesia was being laid out, the only mapmakers were in the neighboring town of
Dyslexia. The minorlevel functionary who sent the contract out to bid neglected take into account
that the dyslectic mapmaker was equally likely to draw his maps with north pointing up, down, left,
or right. Although he always drew the eastwest line at right angles to the northsouth line, he
often got east and west backwards. When his boss realized that the job had gone to the lowest
bidder, who happened to live in Dyslexia, many hours were spent in committee meetings trying to
figure out what to do. The paperwork had been done, the purchase order had been issued, and
bureaucracies being what they are, it would be too expensive and timeconsuming to cancel the
order. Still, nobody had any idea what the mapmaker would deliver. A committee was hastily formed.
The committee fairly quickly decided that there were only eight possible orientations that the
mapmaker could deliver, shown in Figure 1.7. In the best of all possible
worlds, he would deliver a map oriented as shownin the topleft rectangle, with north pointing to
the top of thepage and east to the right, which is what people usually expect. A subcommittee
formed for the task decided to name this the normalorientation.
After the meeting had lasted a few hours and tempers were beginning to fray, it was decided that
the other three variants shown in the top row of Figure 1.7 were probably
acceptable too, because they could be transformed to the normal orientation by placing a pin in the
center of the page and rotating the map around the pin. (You can do this, too, by placing this book
flat on a table and turning it.) Many hours were wasted by tired functionaries putting pins into
various places in the maps shown in the second row of Figure 1.7, but no
matter how fast they twirled them, they couldn't seem to transform them to the normal orientation.
It wasn't until everybody important had given up and gone home that a tired intern, assigned to
clean up the used coffee cups, noticed that the maps in the second row can be transformed into the
normal orientation by holding them up against a light and viewing them from the back. (You can do
this, too, by holding Figure 1.7 up to the light and viewing it from the
back—you'll have to turn it, too, of course.) The writing was backwards too, but it was decided
that if
Leonardo da Vinci (1452–1519) could handle backwards writing in 15th century, then the citizens of
Cartesia, though by no means his intellectual equivalent (probably due to daytime TV), could
probably handle it in the 21st century.
In summary, no matter what orientation we choose for the
$x$
 and
$y$
axes, we can always rotate
the coordinate space around so that
$+x$
points to our right and
$+y$
points up. For our example
of screenspace coordinates, imagine turning the coordinate system upside down and looking at the
screen from behind the monitor. In any case, these rotations do not distort the original shape of
the coordinate system (even though we may be looking at it upside down or reversed). So in one
particular sense, all 2D coordinate systems are “equal.” In Section 1.3.3, we discover
the surprising fact that this is not the case in 3D.
1.2.3Specifying Locations in 2D Using Cartesian Coordinates
A coordinate space is a framework for specifying location precisely. A gentleman of Cartesia could,
if he wished to tell his lady love where to meet him for dinner, for example, consult the map in
Figure 1.4 and say, “Meet you at the corner of East 2nd Street and North 4th
Street.” Notice that he specifies two coordinates, one in the horizontal dimension (East 2nd
Street, listed along the top of the map in Figure 1.4) and one in the vertical
dimension (North 4th Street, listed along the left of the map). If he wished to be concise he could
abbreviate the “East 2nd Street” to “2” and the “North 4th Street” to “4” and say to his
lady love, somewhat cryptically, “Meet you at (
$2,4$
).”
The ordered pair (
$2,4$
) is an example of what are called Cartesian coordinates. In 2D, two
numbers are used to specify a location. (The fact that we use two numbers to describe the location
of a point is the reason it's called twodimensional space. In 3D, we will use three
numbers.) The first coordinate (the
$2$
in our example (
$2,4$
)) is called the
$x$
coordinate, and
the second coordinate (the
$4$
in our example (
$2,4$
)) is called the
$y$
coordinate.
Analogous to the street names in Cartesia, each of the two coordinates specifies which side of the
origin the point is on and how far away the point is from the origin in that direction. More
precisely, each coordinate is the
signed distance (that is, positive in one direction and negative in the other) to one of the
axes, measured along a line parallel to the other axis. Essentially, we use positive coordinates
for east and north streets and negative coordinates for south and west streets. As shown in
Figure 1.8, the
$x$
coordinate designates the signed distance from the point to
the
$y$
axis, measured along a line parallel to the
$x$
axis. Likewise, the
$y$
coordinate
designates the signed distance from the point to the
$x$
axis, measured along a line parallel to
the
$y$
axis.
Figure 1.9 shows several points and their Cartesian coordinates. Notice that the
points to the left of the
$y$
axis have negative
$x$
values, and those to the right of the
$y$
axis
have positive
$x$
values. Likewise, points with positive
$y$
are located above the
$x$
axis, and
points with negative
$y$
are below the
$x$
axis. Also notice that any point can be
specified, not just the points at
grid line intersections. You should study this figure until you are sure that you understand the
pattern.
Let's take a closer look at the grid lines usually shown in a
diagram. Notice that a vertical grid line is composed of points that
all have the same
$x$
coordinate. In other words, a vertical grid
line (actually any vertical line) marks a line of constant
$x$
. Likewise, a horizontal grid line marks a line of constant
$y$
;
all the points on that line have the same
$y$
coordinate. We'll
come back to this idea in a bit when we discuss polar coordinatespaces.
1.33D Cartesian Space
The previous sections have explained how the Cartesian coordinate system works in 2D. Now it's
time to leave the flat 2D world and think about 3D space.
It might seem at first that 3D space is only “50%more complicated” than 2D. After all, it's
just one more dimension, and we already had two. Unfortunately, this is not the case.
For a variety of reasons, 3D space is more than incrementally more difficult than 2D space
for humans to visualize and describe. (One possible reason for this difficulty could be that our
physical world is 3D, whereas illustrations in books and on computer screens are 2D.) It is
frequently the case that a problem that is “easy” to solve in 2D is much more difficult or even
undefined in 3D. Still, many concepts in 2D do extend directly into 3D, and we frequently use 2D to
establish an understanding of a problem and develop a solution, and then extend that solution into
3D.
This section extends 2D Cartesian math into 3D. It is divided into
four major subsections.

Section 1.3.1 begins the extension of 2D into 3D by adding a third axis. The
main concepts introduced are

the
$z$
axis

the
$xy$
,
$xz$
, and
$yz$
planes

Section 1.3.2 describes how to specify the
location of a point in the 3D plane using Cartesian
$(x,y,z)$
coordinates.

Section 1.3.3 introduces the concepts of lefthanded and righthanded 3D
coordinate spaces. The main concepts introduced are

the hand rule, an informal definition for lefthanded and righthanded
coordinate spaces

differences in rotation in lefthanded and righthanded coordinate spaces

how to convert between the two

neither is better than the other, only different

Section 1.3.4 describes some conventions
used in this book.
1.3.1Extra Dimension, Extra Axis
In 3D, we require three axes to establish a coordinate system. The first two axes are called the
$x$
axis and
$y$
axis, just as in 2D. (However, it is not accurate to say that these are the
same as the 2D axes; more on this later.) We call the third axis (predictably) the
$z$
axis.
Usually, we set things up so that all axes are mutually perpendicular, that is, each one is
perpendicular to the others. Figure 1.10 shows an example of a 3D coordinate
space.
As discussed in Section 1.2.2, it is customary in 2D for
$+x$
to point to the right and
$+y$
to point up. (Or sometimes
$+y$
may point down, but in either case, the
$x$
axis is
horizontal and the
$y$
axis is vertical.) These conventions in 2D are fairly standardized. In 3D,
however, the conventions for arrangement of the axes in diagrams and the assignment of the axes
onto physical dimensions (left, right, up, down, forward, back) are not very standardized.
Different authors and fields of study have different conventions. Section 1.3.4
discusses the conventions used in thisbook.
As mentioned earlier, it is not entirely appropriate to say that the
$x$
axis and
$y$
axis in 3D are the “same” as the
$x$
axis and
$y$
axis in 2D. In 3D, any pair of axes defines a plane that
contains the two axes and is perpendicular to the third axis. For
example, the plane containing the
$x$
 and
$y$
axes is the
$xy$
plane, which is perpendicular to the
$z$
axis. Likewise, the
$xz$
plane is perpendicular to the
$y$
axis, and the
$yz$
plane is
perpendicular to the
$x$
axis. We can consider any of these planes
a 2D Cartesian coordinate space in its own right. For example, if
we assign
$+x$
,
$+y$
, and
$+z$
to point right, up, and forward,
respectively, then the 2D coordinate space of the “ground” is the
$xz$
plane, as shown in Figure 1.10.
1.3.2Specifying Locations in 3D
In 3D, points are specified using three numbers,
$x$
,
$y$
, and
$z$
, which give the signed distance
to the
$yz$
,
$xz$
, and
$xy$
planes, respectively. This distance is measured along a line parallel
to the axis. For example, the
$x$
value is the signed distance to the
$yz$
plane, measured along a
line parallel to the
$x$
axis. Don't let this precise definition of how points in 3D are located
confuse you. It is a straightforward extension of the process for 2D, as shown in
Figure 1.11.
1.3.3Lefthanded versus Righthanded Coordinate Spaces
As we discussed in Section 1.2.2, all 2D coordinate systems are “equal” in the sense
that for any two 2D coordinate spaces
$A$
and
$B$
, we can rotate coordinate space
$A$
such that
$+x$
and
$+y$
point in the same direction as they do in coordinate space
$B$
. (We are assuming
perpendicular axes.) Let's examine this idea in more detail.
Figure 1.5 shows the “standard” 2D
coordinate space. Notice that the difference between this
coordinate space and “screen” coordinate space shown
Figure 1.6 is that the
$y$
axis points in opposite
directions. However, imagine rotating Figure 1.6
clockwise 180 degrees so that
$+y$
points up and
$+x$
points to
the left. Now rotate it by “turning the page” and viewing the
diagram from behind. Notice that now the axes are oriented in the
“standard” directions like in Figure 1.5. No
matter how many times we flip an axis, we can always find a way to
rotate things back into the standard orientation.
Let's see how this idea extends into 3D. Examine
Figure 1.10 once more. We stated earlier that
$+z$
points into the page. Does it have to be this way? What if we
made
$+z$
point out of the page? This is certainly allowed, so
let's flip the
$z$
axis.
Now, can we rotate the coordinate system around such that things line up with the original
coordinate system? As it turns out, we cannot. We can rotate things to line up two axes at a time,
but the third axes always points in the wrong direction! (If you have trouble visualizing this,
don't worry. In just a moment we will illustrate this principle in more concrete terms.)
All 3D coordinate spaces are not equal, in the sense that some pairs of coordinate systems cannot
be rotated to line up with each other. There are exactly two distinct types of 3D coordinate
spaces: lefthanded coordinate spaces and righthanded coordinate spaces. If two
coordinate spaces have the same handedness, then they can be rotated such that the axes are
aligned. If they are of opposite handedness, then this is not possible.
What exactly do “lefthanded” and “righthanded” mean? The most intuitive way to identify the
handedness of a particular coordinate system is to use, well, your hands!
With your left hand, make an `L' with your thumb and index finger. Your thumb should be pointing to your right, and your index finger should be pointing
up. Now extend your third finger so it points directly forward. You have just formed a lefthanded coordinate
system. Your thumb, index finger, and third finger point in the
$+x$
,
$+y$
, and
$+z$
directions,
respectively. This is shown in Figure 1.12.
Now perform the same experiment with your right hand. Notice that your index finger still points
up, and your third finger points forward. However, with your right hand, your thumb will point to
the left. This is a righthanded coordinate system. Again, your thumb, index finger, and third
finger point in the
$+x$
,
$+y$
, and
$+z$
directions, respectively. A
righthanded coordinate system
is shown in Figure 1.13.
Try as you might, you cannot rotate your hands into a position such that all three fingers
simultaneously point the same direction on both hands. (Bending your fingers is not allowed.)
Lefthanded and righthanded coordinate systems also differ in the definition of “positive
rotation.” Let's say we a have line in space and we need to rotate about this line by a
specified angle. We call this line an axis of rotation, but don't think that the word
axis implies that we're talking only about one of the cardinal axes (the
$x$
,
$y$
, or
$z$
axis). An axis of rotation can be arbitrarily oriented. Now, if you tell me to “rotate
$30{}^{\mathrm{o}}$
about the axis,” how do I know which way to rotate? We need to agree between us
that one direction of rotation is the positive direction, and the other direction is the negative
direction. The standard way to tell which is which in a lefthanded coordinate system is called
the lefthand rule. First, we must define which way our axis “points.” Of course, the
axis of rotation is theoretically infinite in length, but we still consider it having a positive
and negative end, just like the normal cardinal axes that define our coordinate space. The
lefthand rule works like this: put your left hand in the “thumbs up” position, with your thumb
pointing towards the positive end of the axis of rotation. Positive rotation about the axis of
rotation is in the direction that your fingers are curled. There's a corresponding rule for
righthanded coordinate spaces; both of these rules are illustrated in
Figure 1.14.
As you can see, in a lefthanded coordinate system, positive rotation rotates clockwise when
viewed from the positive end of the axis, and in a righthanded coordinate system, positive
rotation is counterclockwise. Table 1.1 shows what happens when we
apply this general rule to the specific case of the cardinal axes.
When looking towards the origin from… 
Positive rotation 
Negative rotation 
Lefthanded: Clockwise 
Lefthanded: Counterclockwise 
Righthanded: Counterclockwise 
Righthanded: Clockwise 
$+x$

$+y\to +z\to y\to z\to +y$

$+y\to z\to y\to +z\to +y$

$+y$

$+z\to +x\to z\to x\to +z$

$+z\to x\to z\to +x\to +z$

$+z$

$+x\to +y\to x\to y\to +x$

$+x\to y\to x\to +y\to +x$

Table 1.1
Rotation about the cardinal axes in left and righthanded
coordinate systems
Any lefthanded coordinate system can be transformed into a righthanded coordinate system, or vice
versa. The simplest way to do this is by swapping the positive and negative ends of one axis.
Notice that if we flip two axes, it is the same as rotating the coordinate space
$180{}^{\mathrm{o}}$
about the third axis, which does not change the handedness of the coordinate space.
Another way to toggle the handedness of a coordinate system is to exchange two axes.
Both lefthanded and righthanded coordinate systems are perfectly valid, and despite what you
might read in other books, neither is “better” than the other. People in various fields of study
certainly have preferences for one or the other, depending on their backgrounds. For example, some
newer computer graphics literature uses lefthanded coordinate systems, whereas traditional
graphics texts and more mathoriented linear algebra people tend to prefer righthanded coordinate
systems. Of course, these are gross generalizations, so always check to see what coordinate system
is being used. The bottom line, however, is that in many cases it's just a matter of a negative
sign in the
$z$
coordinate. So, appealing to the first law of computer graphics in
Section 1.1, if you apply a tool, technique, or resource from another book, web page, or
article and it doesn't look right, try flipping the sign on the
$z$
axis.
1.3.4Some Important Conventions Used in This Book
When designing a 3D virtual world, several design decisions have to be made beforehand, such as
lefthanded or righthanded coordinate system, which direction is
$+y$
, and so forth. The map
makers from Dyslexia had to choose from among eight different ways to assign the axes in 2D (see
Figure 1.7). In 3D, we have a total of 48 different combinations to choose
from; 24 of these combinations are lefthanded, and 24 are righthanded.
(Exercise 3 asks you to list all of them.)
Different situations can call for different conventions, in the sense that certain tasks can be
easier if you adopt the right conventions. Usually, however, it is not a major deal as long as you
establish the conventions early in your design process and stick to them. (In fact, the choice is
most likely thrust upon you by the engine or framework you are using, because very few people start
from scratch these days.) All of the basic principles discussed in this book are applicable
regardless of the conventions used. For the most part, all of the equations and techniques given
are applicable regardless of convention, as well. However, in
some cases there are some slight, but critical, differences in application dealing with lefthanded
versus righthanded coordinate spaces. When those differences arise, we will point them out.
We use a lefthanded coordinate system in this book. The
$+x$
,
$+y$
, and
$+z$
directions point
right, up, and forward, respectively, as shown in Figure 1.15. In situations where
“right” and “forward” are not appropriate terms (for example, when we discuss the world
coordinate space), we assign
$+x$
to “east” and
$+z$
to “north.”
1.4Odds and Ends
In this book, we spend a lot of time focusing on some crucial material that is often relegated to a
terse presentation tucked away in an appendix in the books that consider this material a
prerequisite. We, too, must assume a nonzero level of mathematical knowledge from the reader, or
else every book would get no further than a review of first principles, and so we also have our
terse presentation of some prerequisites. In this section we present a few bits of mathematical
knowledge with which most readers are probably familiar, but might need a quick refresher.
1.4.1Summation and Product Notation
Summation notation is a shorthand way to write the sum of a list of things. It's sort of
like a mathematical for loop. Let's look at an example:
Summation notation
$$\sum _{i=1}^{6}{a}_{i}={a}_{1}+{a}_{2}+{a}_{3}+{a}_{4}+{a}_{5}+{a}_{6}.$$
The variable
$i$
is known as the index variable. The expressions above and below the
summation symbol tell us how many times to execute our “loop” and what values to use for
$i$
during each iteration. In this case,
$i$
will count from 1 to 6. To “execute” our loop, we
iterate the index through all the values specified by the control conditions. For each iteration,
we evaluate the expression on the righthand side of the summation notation (substituting the
appropriate value for the index variable), and add this to our sum.
Summation notation is also known as sigma notation because that coollooking symbol that
looks like an E is the capital version of the Greek letter sigma.
A similar notation is used when we are taking the product of a series of values, only we use the
symbol
$\mathrm{\Pi}$
, which is the capital version of the letter
$\pi $
:
Product notation
$$\prod _{i=1}^{n}{a}_{i}={a}_{1}\times {a}_{2}\times \cdots \times {a}_{n1}\times {a}_{n}.$$
1.4.2Interval Notation
Several times in this book, we refer to a subset of the real number line using interval
notation. The notation
$[a,b]$
means, “the portion of the number line from
$a$
to
$b$
.” Or,
more formally, we could read
$[a,b]$
as “all numbers
$x$
such that
$a\le x\le b$
.” Notice that
this is a closed interval, meaning that the endpoints
$a$
and
$b$
are included in the
interval. An open interval is one in which the endpoints are excluded. It is denoted using
parentheses instead of square brackets:
$(a,b)$
. This interval contains all
$x$
such that
$a<x<b$
. Sometimes a closed interval is called inclusive and an open interval called
exclusive.
Occasionally, we encounter halfopen intervals, which include one endpoint but exclude the
other. These are denoted with a lopsided notation such as
$[a,b)$
or
$(a,b]$
, with the square bracket being placed
next to the endpoint that is included. By convention, if an endpoint is infinite, we consider that
end to be open. For example, the set of all nonnegative numbers is
$[0,\mathrm{\infty})$
.
Notice that the notation
$(x,y)$
could refer to an open interval or a 2D point. Likewise,
$[x,y]$
could be a closed interval or a 2D vector (discussed in the next chapter). The context will
always make clear which is the case.
1.4.3Angles, Degrees, and Radians
An angle measures an amount of rotation in the plane. Variables representing angles are often
assigned the Greek letter
$\theta $
. The most important units of measure used to specify angles are degrees
(°) and radians (rad).
Humans usually measure angles using degrees. One degree measures 1/360th of a revolution,
so
$360{}^{\mathrm{o}}$
represents a complete revolution.
Mathematicians, however, prefer to measure angles in radians, which is a unit of measure
based on the properties of a circle. When we specify the angle between two rays in radians, we are
actually measuring the length of the intercepted arc of a unit circle (a circle centered at the origin with radius 1), as shown in
Figure 1.16.
The circumference of a unit circle is
$2\pi $
, with
$\pi $
approximately equal to
$3.14159265359$
.
Therefore,
$2\pi $
radians represents a complete revolution.
Since
$360{}^{\mathrm{o}}=2\pi \text{}\mathrm{r}\mathrm{a}\mathrm{d}$
,
$180{}^{\mathrm{o}}=\pi \text{}\mathrm{r}\mathrm{a}\mathrm{d}$
. To convert an angle from
radians to degrees, we multiply by
$180/\pi \approx 57.29578$
, and to convert an angle from degrees
to radians, we multiply by
$\pi /180\approx 0.01745329$
. Thus,
Converting between radians and degrees
$$\begin{array}{rll}1\text{}\mathrm{r}\mathrm{a}\mathrm{d}=& \left(180/\pi \right){}^{\mathrm{o}}& \approx 57.29578{}^{\mathrm{o}},\\ 1{}^{\mathrm{o}}=& \left(\pi /180\right)\text{}\mathrm{r}\mathrm{a}\mathrm{d}& \approx 0.01745329\text{}\mathrm{r}\mathrm{a}\mathrm{d}.\end{array}$$
In the next section, Table 1.2 will list several angles in both degree and radian
format.
1.4.4Trig Functions
There are many ways to define the elementary trig functions. In this section, we define them using
the unit circle. In two dimensions, if we begin
with a unit ray pointing towards
$+x$
, and then rotate this ray counterclockwise by an angle
$\theta $
, we have drawn the angle in the standard position. (If the angle is negative,
rotate the ray in the other direction.) This is illustrated in
Figure 1.17.
The
$(x,y)$
coordinates of the endpoint of a ray thus rotated have special properties and are so
significant mathematically that they have been assigned special functions, known as the
cosine and sine of the angle:
Defining sine and cosine using the unit circle
$$\begin{array}{rlrl}\mathrm{cos}\theta & =x,& \mathrm{sin}\theta & =y.\end{array}$$
You can easily remember which is which because they are in alphabetical order:
$x$
comes before
$y$
, and
$\mathrm{cos}$
comes before
$\mathrm{sin}$
.
The secant, cosecant, tangent, and cotangent are also useful trig
functions. They can be defined in terms of the the sine and cosine:
$$\begin{array}{rlrl}\mathrm{sec}\theta & =\frac{1}{\mathrm{cos}\theta},& \mathrm{tan}\theta & =\frac{\mathrm{sin}\theta}{\mathrm{cos}\theta},\\ \mathrm{csc}\theta & =\frac{1}{\mathrm{sin}\theta},& \mathrm{cot}\theta & =\frac{1}{\mathrm{tan}\theta}=\frac{\mathrm{cos}\theta}{\mathrm{sin}\theta}.\end{array}$$
If we form a right triangle using the rotated ray as the hypotenuse (the side opposite the right
angle), we see that
$x$
and
$y$
give the lengths of the legs (those sides that form the right
angle). The length of the adjacent leg is
$x$
, and the length of the opposite leg is
$y$
, with the
terms “adjacent” and “opposite” interpreted relative to the angle
$\theta $
. Again, alphabetical
order is a useful memory aid: “adjacent” and “opposite” are in the same order as the
corresponding “cosine” and “sine.” Let the abbreviations
$\mathit{h}\mathit{y}\mathit{p}$
,
$\mathit{a}\mathit{d}\mathit{j}$
, and
$\mathit{o}\mathit{p}\mathit{p}$
refer to
the lengths of the hypotenuse, adjacent leg, and opposite leg, respectively, as shown in
Figure 1.18.
The primary trig functions are defined by the following ratios:
$$\begin{array}{rlrlrl}\mathrm{cos}\theta & =\frac{\mathit{a}\mathit{d}\mathit{j}}{\mathit{h}\mathit{y}\mathit{p}},& \mathrm{sin}\theta & =\frac{\mathit{o}\mathit{p}\mathit{p}}{\mathit{h}\mathit{y}\mathit{p}},& \mathrm{tan}\theta & =\frac{\mathit{o}\mathit{p}\mathit{p}}{\mathit{a}\mathit{d}\mathit{j}},\\ \\ \mathrm{sec}\theta & =\frac{\mathit{h}\mathit{y}\mathit{p}}{\mathit{a}\mathit{d}\mathit{j}},& \mathrm{csc}\theta & =\frac{\mathit{h}\mathit{y}\mathit{p}}{\mathit{o}\mathit{p}\mathit{p}},& \mathrm{cot}\theta & =\frac{\mathit{a}\mathit{d}\mathit{j}}{\mathit{o}\mathit{p}\mathit{p}}.\end{array}$$
Because of the properties of similar triangles, the above equations apply even when the hypotenuse
is not of unit length. However, they do not apply when
$\theta $
is obtuse, since we cannot form a
right triangle with an obtuse interior angle. But by showing the angle in standard position and allowing the rotated ray to be of any length
$r$
(Figure 1.19), we can express the ratios using
$x$
,
$y$
, and
$r$
:
$$\begin{array}{rlrlrl}\mathrm{cos}\theta & =x/r,& \mathrm{sin}\theta & =y/r,& \mathrm{tan}\theta & =y/x,\\ \\ \mathrm{sec}\theta & =r/x,& \mathrm{csc}\theta & =r/y,& \mathrm{cot}\theta & =x/y.\end{array}$$
Table 1.2 shows several different angles, expressed in degrees and radians, and the
values of their principal trig functions.
$\begin{array}{rlrrcccc}\theta {}^{\mathrm{o}}& \theta \text{}\mathrm{r}\mathrm{a}\mathrm{d}& \mathrm{cos}\theta & \mathrm{sin}\theta & \mathrm{tan}\theta & \mathrm{sec}\theta & \mathrm{csc}\theta & \mathrm{cot}\theta \\ 0& 0& 1& 0& 0& 1& \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{f}& \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{f}\\ 30& \frac{\pi}{6}\approx 0.5236& \frac{\sqrt{3}}{2}& \frac{1}{2}& \frac{\sqrt{3}}{3}& \frac{2\sqrt{3}}{3}& 2& \sqrt{3}\\ 45& \frac{\pi}{4}\approx 0.7854& \frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}& 1& \sqrt{2}& \sqrt{2}& 1\\ 60& \frac{\pi}{3}\approx 1.0472& \frac{1}{2}& \frac{\sqrt{3}}{2}& \sqrt{3}& 2& \frac{2\sqrt{3}}{3}& \frac{\sqrt{3}}{3}\\ 90& \frac{\pi}{2}\approx 1.5708& 0& 1& \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{f}& \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{f}& 1& 0\\ 120& \frac{2\pi}{3}\approx 2.0944& \frac{1}{2}& \frac{\sqrt{3}}{2}& \sqrt{3}& 2& \frac{2\sqrt{3}}{3}& \frac{\sqrt{3}}{3}\\ 135& \frac{3\pi}{4}\approx 2.3562& \frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}& 1& \sqrt{2}& \sqrt{2}& 1\\ 150& \frac{5\pi}{6}\approx 2.6180& \frac{\sqrt{3}}{2}& \frac{1}{2}& \frac{\sqrt{3}}{3}& \frac{2\sqrt{3}}{3}& 2& \sqrt{3}\\ 180& \pi \approx 3.1416& 1& 0& 0& 1& \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{f}& \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{f}\\ 210& \frac{7\pi}{6}\approx 3.6652& \frac{\sqrt{3}}{2}& \frac{1}{2}& \frac{\sqrt{3}}{3}& \frac{2\sqrt{3}}{3}& 2& \sqrt{3}\\ 225& \frac{5\pi}{4}\approx 3.9270& \frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}& 1& \sqrt{2}& \sqrt{2}& 1\\ 240& \frac{4\pi}{3}\approx 4.1888& \frac{1}{2}& \frac{\sqrt{3}}{2}& \sqrt{3}& 2& \frac{2\sqrt{3}}{3}& \frac{\sqrt{3}}{3}\\ 270& \frac{3\pi}{2}\approx 4.7124& 0& 1& \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{f}& \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{f}& 1& 0\\ 300& \frac{5\pi}{3}\approx 5.2360& \frac{1}{2}& \frac{\sqrt{3}}{2}& \sqrt{3}& 2& \frac{2\sqrt{3}}{3}& \frac{\sqrt{3}}{3}\\ 315& \frac{7\pi}{4}\approx 5.4978& \frac{\sqrt{2}}{2}& \frac{\sqrt{2}}{2}& 1& \sqrt{2}& \sqrt{2}& 1\\ 330& \frac{11\pi}{6}\approx 5.7596& \frac{\sqrt{3}}{2}& \frac{1}{2}& \frac{\sqrt{3}}{3}& \frac{2\sqrt{3}}{3}& 2& \sqrt{3}\\ 360& 2\pi \approx 6.2832& 1& 0& 0& 1& \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{f}& \mathrm{u}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{f}\end{array}$
Table 1.2Common angles in degrees and radians, and the values of the principal trig functions
1.4.5Trig Identities
In this section we present a number of basic relationships between the trig functions. Because we
assume in this book that the reader has some prior exposure to trigonometry, we do not develop or
prove these theorems. The proofs can be found online or in any trigonometry textbook.
A number of identities can be derived based on the symmetry of the unit circle:
Basic identities related to symmetry
$$\begin{array}{rlrlrl}\mathrm{sin}(\theta )=& \mathrm{sin}\theta ,& \mathrm{cos}(\theta )=& \mathrm{cos}\theta ,& \mathrm{tan}(\theta )=& \mathrm{tan}\theta ,\\ \mathrm{sin}(\frac{\pi}{2}\theta )=& \mathrm{cos}\theta ,& \mathrm{cos}(\frac{\pi}{2}\theta )=& \mathrm{sin}\theta ,& \mathrm{tan}(\frac{\pi}{2}\theta )=& \mathrm{cot}\theta .\end{array}$$
Perhaps the most famous and basic identity concerning the right triangle, one that most readers
learned in their primary education, is the
Pythagorean theorem. It says that the sum of the squares of the two legs of a right
triangle is equal to the square of the hypotenuse. Or, more famously, as shown in
Figure 1.20,
Pythagorean theorem
$${a}^{2}+{b}^{2}={c}^{2}.$$
By applying the Pythagorean theorem to the unit circle, one can deduce the identities
Pythagorean identities
$$\begin{array}{rlrlrl}{\mathrm{sin}}^{2}\theta +{\mathrm{cos}}^{2}\theta & =1,& 1+{\mathrm{tan}}^{2}\theta & ={\mathrm{sec}}^{2}\theta ,& 1+{\mathrm{cot}}^{2}\theta & ={\mathrm{csc}}^{2}\theta .\end{array}$$
The following identities involve taking a trig function on the sum or difference of two angles:
Sum and difference identities
$$\begin{array}{rl}\mathrm{sin}(a+b)& =\mathrm{sin}a\mathrm{cos}b+\mathrm{cos}a\mathrm{sin}b,\\ \text{(1.1)}& \mathrm{sin}(ab)& =\mathrm{sin}a\mathrm{cos}b\mathrm{cos}a\mathrm{sin}b,\mathrm{cos}(a+b)& =\mathrm{cos}a\mathrm{cos}b\mathrm{sin}a\mathrm{sin}b,\\ \mathrm{cos}(ab)& =\mathrm{cos}a\mathrm{cos}b+\mathrm{sin}a\mathrm{sin}b,\\ \mathrm{tan}(a+b)& =\frac{\mathrm{tan}a+\mathrm{tan}b}{1\mathrm{tan}a\mathrm{tan}b},\\ \mathrm{tan}(ab)& =\frac{\mathrm{tan}a\mathrm{tan}b}{1+\mathrm{tan}a\mathrm{tan}b}.\end{array}$$
If we apply the sum identities to the special case where
$a$
and
$b$
are the same, we get the
following double angle identities:
Double angle identities
$$\begin{array}{rl}\mathrm{sin}2\theta & =2\mathrm{sin}\theta \mathrm{cos}\theta ,\\ \mathrm{cos}2\theta & ={\mathrm{cos}}^{2}\theta {\mathrm{sin}}^{2}\theta =2{\mathrm{cos}}^{2}\theta 1=12{\mathrm{sin}}^{2}\theta ,\\ \mathrm{tan}2\theta & =\frac{2\mathrm{tan}\theta}{1{\mathrm{tan}}^{2}\theta}.\end{array}$$
We often need to solve for an unknown side length or angle in a triangle, in terms of the known
side lengths or angles. For these types of problems the law of sines and law of
cosines are helpful. The formula to use will depend on which values are known and which value is
unknown. Figure 1.21 illustrates the notation and shows that these
identities hold for any triangle, not just right triangles:
Law of sines
$$\frac{\mathrm{sin}A}{a}=\frac{\mathrm{sin}B}{b}=\frac{\mathrm{sin}C}{c},$$
Law of cosines
$$\begin{array}{rl}{a}^{2}& ={b}^{2}+{c}^{2}2bc\mathrm{cos}A,\\ {b}^{2}& ={a}^{2}+{c}^{2}2ac\mathrm{cos}B,\\ {c}^{2}& ={a}^{2}+{b}^{2}2ab\mathrm{cos}C.\end{array}$$
Exercises

Give the coordinates of the following points. Assume the standard 2D conventions. The
darker grid lines represent one unit.

Give the coordinates of the following points:

List the 48 different possible ways that the 3D axes may be
assigned to the directions “north,” “east,” and “up.”
Identify which of these combinations are lefthanded, and
which are righthanded.

In the popular modeling program
3DS Max, the default
orientation of the axes is for
$+x$
to point right/east,
$+y$
to point forward/north, and
$+z$
to point up.
 (a)Is this a left or righthanded coordinate space?

(b)How would we convert 3D coordinates from the coordinate system used by 3DS
Max into points we could use with our coordinate conventions discussed in
Section 1.3.4?
 (c)What about converting from our conventions to the 3DS Max conventions?

A common convention in aerospace is that
$+x$
points forward/north,
$+y$
points
right/east, and
$z$
points down.
 (a)Is this a left or righthanded coordinate space?

(b)How would we convert 3D coordinates from these aerospace conventions into our
conventions?
 (c)What about converting from our conventions to the aerospace conventions?

In a lefthanded coordinate system:

(a)when looking from the positive end of an axis of rotation, is positive
rotation clockwise (CW) or counterclockwise (CCW)?

(b)when looking from the negative end of an axis of rotation, is positive
rotation CW or CCW?
In a righthanded coordinate system:

(c)when looking from the positive end of an axis of rotation, is positive
rotation CW or CCW?

(d)when looking from the negative end of an axis of rotation, is positive
rotation CW or CCW?

Compute the following:
(a)
$\sum _{i=1}^{5}i$
(b)
$\sum _{i=1}^{5}2i$
(c)
$\prod _{i=1}^{5}2i$
(d)
$\prod _{i=0}^{4}7(i+1)$
(e)
$\sum _{i=1}^{100}i$

Convert from degrees to radians:
(a)
$30{}^{\mathrm{o}}$

(b)
$45{}^{\mathrm{o}}$

(c)
$60{}^{\mathrm{o}}$

(d)
$90{}^{\mathrm{o}}$

(e)
$180{}^{\mathrm{o}}$

(f)
$225{}^{\mathrm{o}}$

(g)
$270{}^{\mathrm{o}}$

(h)
$167.5{}^{\mathrm{o}}$

(i)
$527{}^{\mathrm{o}}$

(j)
$1080{}^{\mathrm{o}}$


Convert from radians to degrees:
(a)
$\pi /6$

(b)
$2\pi /3$

(c)
$3\pi /2$

(d)
$4\pi /3$

(e)
$2\pi $

(f)
$\pi /180$

(g)
$\pi /18$

(h)
$5\pi $

(i)
$10\pi $

(j)
$\pi /5$


In The Wizard of Oz, the scarecrow receives his degree
from the wizard and blurts out this mangled version of the Pythagorean theorem:
The sum of the square roots of any two sides of an isosceles triangle is
equal to the square root of the remaining side.
Apparently the scarecrow's degree wasn't worth very much, since
this “proof that he had a brain” is actually wrong in at least
two ways.
What should the scarecrow have said?

Confirm the following:
 (a)
$(\mathrm{sin}(\alpha )/\mathrm{csc}(\alpha ))+(\mathrm{cos}(\alpha )/\mathrm{sec}(\alpha ))=1$
 (b)
$({\mathrm{sec}}^{2}(\theta )1)/{\mathrm{sec}}^{2}(\theta )={\mathrm{sin}}^{2}(\theta )$
 (c)
$1+{\mathrm{cot}}^{2}(t)={\mathrm{csc}}^{2}(t)$
 (d)
$\mathrm{cos}(\varphi )(\mathrm{tan}(\varphi )+\mathrm{cot}(\varphi ))=\mathrm{csc}(\varphi )$
People, places, science, odds and ends, and things you should have
learned in school had you been paying attention.
— Categories from Michael Feldman's
weekend radio show Whaddya know?