FUNCTIONS AND PLANES CYLINDRICAL SPHERICAL AND POLAR COORDINATES

FUNCTIONS AND PLANES, CYLINDRICAL, SPHERICAL AND POLAR COORDINATES

COORDINATES SYSTEMS 2 -2

RECTANGULAR COORDINATE SYSTEM 2 -3

DIFFERENTIAL VOLUME ELEMENT

VECTOR EXPRESSIONS IN RECTANGULAR COORDINATES General Vector, B: Magnitude of B: Unit Vector in the Direction of B:

CYLINDRICAL COORDINATE SYSTEMS

CYLINDRICAL COORDINATE SYSTEMS 2 -7

CYLINDRICAL COORDINATE SYSTEMS EXAMPLE: 2 -8

CYLINDRICAL COORDINATE SYSTEMS 1 -9

DIFFERENTIAL VOLUME IN CYLINDRICAL COORDINATES d. V = d d dz 2 -10

POINT TRANSFORMATIONS IN CYLINDRICAL COORDINATES

THE RELATIONSHIP BETWEEN UNITS 1 -12

1 -13

z Transformations between cylindrical and Cartesian From Cartesian to cylindrical y x From cylindrical to Cartesian

1 -15

1 -16

Spherical Coordinates

Spherical Coordinates 2 -18

Spherical Coordinates 2 -19

Spherical Coordinates 2 -20

Spherical Coordinates 2 -21

SPHERICAL COORDINATES Point P has coordinates Specified by P(r ) 2 -22

DIFFERENTIAL VOLUME IN SPHERICAL COORDINATES d. V = r 2 sin drd d 2 -23

Transformations between spherical and Cartesian z From Cartesian to spherical y x From spherical to Cartesian

DIFFERENTIAL ELEMENTS IN RECTANGULAR COORDINATE SYSTEMS 2 -25

DIFFERENTIAL ELEMENTS IN CYLINDRICAL COORDINATE SYSTEMS 2 -26

DIFFERENTIAL ELEMENTS IN SPHERICAL COORDINATE SYSTEMS 2 -27

Example A point P with Cartesian coordinates (2, − 2, 1) has spherical coordinates 1 -28

Example A point P with spherical coordinates (4, π/3 , 3π/4) has Cartesian coordinate 1 -29

Example: Convert the point (4, π/4 , π/6) from spherical to rectangular coordinates Since ρ = 4, θ = π/4, and φ = π/6, x = ρ sin φ cos θ = 4. sin π/6. cos π/4 = 4. ½. (1/√ 2) = √ 2, y = r sin φ sin θ = 4. sin π/6. sin π/4 = 4. 1/2. (1/√ 2) =√ 2, z = ρ cos φ = 4. cos π/6 = 4. (√ 3/2) = 2√ 3. Thus, the point is (√ 2, 2/√ 3) in rectangular coordinates. 1 -30

Example: Convert the point (1, − √ 2) from rectangular to spherical coordinates. 1 -31

1 -32

The first equation gives 1 -33

EXAMPLE Given point P(— 2, 6, 3) and vector A = yax + (x + z)ay, express P and A in cylindrical and spherical coordinates. Solution: At point P: x = - 2 , y = 6, z = 3. Hence, 2 -34

1 -35

1 -36

1 -37

1 -38

Using Polar Coordinates Graphing and converting polar and rectangular coordinates

3 D Polar Space Chapter 7 Notes 3 D Math Primer for Graphics & Game Dev 40

CYLINDRICAL COORDINATES A natural extension of the 2 d polar coordinates are cylindrical coordinates, since they just add a height value out of the xy-plane. For completeness here they are: 1 -41

3 D Polar Space There are two kinds in common use: 1. Cylindrical coordinates – 1 angle and 2 distances 2. Spherical coordinates – 2 angles and 1 distance Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 42

3 D Cylindrical Space To locate the point described by cylindrical coordinates (r, θ, z), start by processing r and θ just like we would for 2 D polar coordinates, and then move up or down the z axis by z. Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 43

3 D Spherical Coordinates • As with 2 D polar coordinates, 3 D spherical coordinates also work by defining a direction and distance. • The only difference is that in 3 D it takes two angles to define a direction. • There are two polar axes in 3 D spherical space. 1. The first axis is horizontal and corresponds to the polar axis in 2 D polar coordinates or +x in our 3 D Cartesian conventions. 2. The other axis is vertical, corresponding to +y in our 3 D Cartesian conventions. Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 44

B utterflies are among the most celebrated of all insects. It’s hard not to notice their beautiful colors and graceful flight. Their symmetry can be explored with trigonometric functions and a system for plotting points called the polar coordinate system. In many cases, polar coordinates are simpler and easier to use than rectangular coordinates.

You are familiar with plotting with a rectangular coordinate system. We are going to look at a new coordinate system called the polar coordinate system.

The center of the graph is called the pole. Angles are measured from the positive x axis. Points are represented by a radius and an angle radius (r, ) To plot the point First find the angle Then move out along the terminal side 5

The polar coordinate system is formed by fixing a point, O, which is the pole (or origin). The polar axis is the ray constructed from O. Each point P in the plane can be assigned polar coordinates (r, ). e c n ta P = (r, ) is d d cte O r ire d = = directed angle Pole (Origin) Polar axis r is the directed distance from O to P. is the directed angle (counterclockwise) from the polar axis to OP. 48

Polar Coordinate Space • Recall that 2 D Cartesian coordinate space has an origin and two axes that pass through the origin. • A 2 D polar coordinate space also has an origin (known as the pole), which has the same basic purpose: it defines the center of the coordinate space. • A polar coordinate space only has one axis, sometimes called the polar axis, which is usually depicted as a ray from the origin. • It is customary in math literature for the polar axis to point to the right in diagrams, and thus it corresponds to the +x axis in a Cartesian system. • It's often convenient to use different conventions than this, as we'll discuss later in this lecture. Until then, we’ll use the traditional conventions of the math literature. Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 49

Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 50

Polar Coordinates • In Cartesian coordinates we described a 2 D point using the using two signed distances, x and y. • Polar coordinates use a distance and an angle. • By convention, the distance is usually called r (which is short for radius) and the angle is usually called θ. The polar coordinate pair (r, θ) species a point in 2 D space as follows: 1. Start at the origin, facing in the direction of the polar axis, and rotate by angle θ. Positive values of θ are usually interpreted to mean counterclockwise rotation, with negative values indicating clockwise rotation. 2. Now move forward from the origin a distance of r units. Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 51

Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 52

Examples Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 53

Polar Diagrams • The grid circles show lines of constant r. • The straight grid lines that pass through the origin show lines of constant θ, consisting of points that are the same direction from the origin. Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 54

Angular Measurement • It really doesn't matter whether you use degrees or radians (or grads, mils, minutes, signs, sextants, or Furmans) to measure angles, so long as you keep it straight. • In the text of our book we almost always give specific angular measurements in degrees and use the ° symbol after the number. • We do this because we are human beings, and most humans who are not math professors find it easier to deal with whole numbers rather than fractions of π. • Indeed, the choice of the number 360 was specifically designed to make fractions avoidable in many common cases. • However, computers prefer to work with angles expressed using radians, and so the code snippets in our book use radians rather than degrees. Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 55

Some Ponderable Questions 1. Can the radial distance r ever be negative? 2. Can θ ever go outside of – 180°≤ θ ≤ 180°? 3. The value of the angle directly west of the origin (i. e. for points where x < 0 and y = 0 using Cartesian coordinates) is ambiguous. Is θ equal to +180° or – 180° for these points? 4. The polar coordinates for the origin itself are also ambiguous. Clearly r = 0, but what value of θ should we use? Wouldn't any value work? Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 56

Aliasing • The answer to all of these questions is “yes”. • In fact, for any given point, there are infinitely many polar coordinate pairs that can be used to describe that point. • This phenomenon is known as aliasing. • Two coordinate pairs are said to be aliases of each other if they have different numeric values but refer to the same point in space. • Notice that aliasing doesn't happen in Cartesian space. Each point in space is assigned exactly one (x, y) coordinate pair. • A given point in polar space corresponds to many coordinate pairs, but a coordinate pair unambiguously designates exactly one point. Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 57

Creating Aliases • One way to create an alias for a point (r, θ) is to add a multiple of 360° to θ. Thus (r, θ) and (r, θ + k 360°) describe the same point, where k is an integer. • We can also generate an alias by adding 180° to θ and negating r; which means we face the other direction, but we displace by the opposite amount. • In general, for any point (r, θ) other than the origin, all of the polar coordinates that are aliases for (r, θ) be expressed as: Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 58

Canonical Polar Coordinates A polar coordinate pair (r, θ) is in canonical form if all of the following are true: Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 59

Algorithm to Make (r, θ) Canonical 1. 2. 3. 4. If r = 0, then assign θ = 0. If r < 0, then negate r, and add 180° to θ. If θ ≤ 180°, then add 360° until θ > – 180° If θ > 180°, then subtract 360° until θ ≤ 180°. Chapter 1 Notes 3 D Math Primer for Graphics & Game Dev 60

Graphing Polar Coordinates A The grid at the left is a polar grid. The typical angles of 30 o, 45 o, 90 o, … are shown on the graph along with circles of radius 1, 2, 3, 4, and 5 units. Points in polar form are given as (r, ) where r is the radius to the point and is the angle of the point. On one of your polar graphs, plot the point (3, 90 o)? The point on the graph labeled A is correct.

A negative angle would be measured clockwise like usual. To plot a point with a negative radius, find the terminal side of the angle but then measure from the pole in the negative direction of the terminal side.

Polar coordinates can also be given with the angle in degrees. 120 90 (8, 210°) 60 45 135 30 150 180 0 (6, -120°) (-5, 300°) 330 210 315 225 240 300 270 (-3, 540°) (540°-360°=180°)

Polar coordinates can also be given with the angle in degrees. 120 90 (8, 210°) 60 45 135 30 150 180 (6, -120°), -120+360=240 0 (-5, 300°), 300 -180=120 330 210 315 225 240 300 270 (-3, 540°), 540 -360 -180=0

Graphing Polar Coordinates Now, try graphing A B C . Did you get point B? Polar points have a new aspect. A radius can be negative! A negative radius means to go in the exact opposite direction of the angle. To graph (-4, 240 o), find 240 o and move 4 units in the opposite direction. The opposite direction is always a 180 o difference. Point C is at (-4, 240 o). This point could also be labeled as (4, 60 o).

Graphing Polar Coordinates A B C How would you write point A with a negative radius? A correct answer would be (-3, 270 o) or (-3, -90 o). In fact, there an infinite number of ways to label a single polar point. Is (3, 450 o) the same point? Don’t forget, you can also use radian angles as well as angles in degrees. On your own, find at least 4 different polar coordinates for point B.

Plotting Points The point lies two units from the pole on the terminal side of the angle 1 2 3 0 3 units from the pole Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 67

There are many ways to represent the point additional ways to represent the 1 2 3 0 point Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 68

Let's plot the following points: Notice unlike in the rectangular coordinate system, there are many ways to list the same point.

Converting from Rectangular to Polar Find the polar form for the rectangular point (4, 3) r 4 3 To find the polar coordinate, we must calculate the radius and angle to the given point. We can use our knowledge of right triangle trigonometry to find the radius and angle. r 2 = 3 2 + 4 2 r 2 = 25 r=5 tan = ¾ = tan-1(¾) = 36. 87 o or 0. 64 rad The polar form of the rectangular point (4, 3) is (5, 36. 87 o)

Converting from Rectangular to Polar In general, the rectangular point (x, y) is converted to polar form (r, θ) by: 1. Finding the radius (x, y) r x y r 2 = x 2 + y 2 2. Finding the angle tan = y/x or = tan-1(y/x) Recall that some angles require the angle to be converted to the appropriate quadrant.

Converting from Rectangular to Polar On your own, find polar form for the point (-2, 3) r 2 = (-2)2 + 32 r 2 = 4 + 9 r 2 = 13 r= However, the angle must be in the second quadrant, so we add 180 o to the answer and get an angle of 123. 70 o. The polar form is ( , 123. 70 o)

Converting from Polar to Rectanglar Convert the polar point (4, 30 o) to rectangular coordinates. We are given the radius of 4 and angle of 30 o. Find the values of x and y. 4 30 o y Using trig to find the values of x and y, we know that cos = x/r or x = r cos . Also, sin = y/r or y = r sin . x The point in rectangular form is:

Converting from Polar to Rectanglar On your own, convert (3, 5π/3) to rectangular coordinates. We are given the radius of 3 and angle of 5π/3 or 300 o. Find the values of x and y. -60 o The point in rectangular form is:

The relationship between rectangular and polar coordinates is as follows. y The point (x, y) lies on a circle of radius r, therefore, r 2 = x 2 + y 2. (x, y) (r, ) r Definitions of trigonometric functions y Pole (Origin) x x Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 75

Coordinate Conversion (Pythagorean Identity) Example: Convert the point into rectangular coordinates. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 76

Example: Convert the point (1, 1) into polar coordinates. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 77

Let's take a point in the rectangular coordinate system and convert it to the polar coordinate system. (3, 4) r 4 Based on the trig you know can you see how to find r and ? 3 r=5 We'll find in radians polar coordinates are: (5, 0. 93)

Let's generalize this to find formulas for converting from rectangular to polar coordinates. (x, y) r x y

Now let's go the other way, from polar to rectangular coordinates. Based on the trig you know can you see how to find x and y? 4 y x rectangular coordinates are:

Let's generalize the conversion from polar to rectangular coordinates. r y x

Convert the rectangular coordinate system equation to a polar coordinate system equation. Here each r unit is 1/2 and we went out 3 and did all angles. Before we do the conversion let's look at the graph. r must be 3 but there is no restriction on so consider all values.

Convert the rectangular coordinate system equation to a polar coordinate system equation. What are the polar conversions we found for x and y? substitute in for x and y We wouldn't recognize what this equation looked like in polar coordinates but looking at the rectangular equation we'd know it was a parabola.

When trying to figure out the graphs of polar equations we can convert them to rectangular equations particularly if we recognize the graph in rectangular coordinates. We could square both sides Now use our conversion: We recognize this as a circle with center at (0, 0) and a radius of 7. On polar graph paper it will centered at the origin and out 7

Let's try another: Take the tangent of both sides To graph on a polar plot we'd go to where and make a line. Now use our conversion: Multiply both sides by x We recognize this as a line with slope square root of 3.

Let's try another: Now use our conversion: We recognize this as a horizontal line 5 units below the origin (or on a polar plot below the pole)

Example: Convert the polar equation. into a rectangular Polar form Multiply each side by r. Substitute rectangular coordinates. Equation of a circle with center (0, 2) and radius of 2 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 88

Rectangular and Polar Equations in rectangular form use variables (x, y), while equations in polar form use variables (r, ) where is an angle. Converting from one form to another involves changing the variables from one form to the other. We have already used all of the conversions which are necessary. Converting Polar to Rectangular cos = x/r sin = y/r tan = y/x r 2 = x 2 + y 2 Converting Rectanglar to Polar x = r cos y = r sin x 2 + y 2 = r 2

Convert Rectangular Equations to Polar Equations The goal is to change all x’s and y’s to r’s and ’s. When possible, solve for r. Example 1: Convert x 2 + y 2 = 16 to polar form. Since x 2 + y 2 = r 2, substitute into the equation. r 2 = 16 Simplify. r=4 r = 4 is the equivalent polar equation to x 2 + y 2 = 16

Example: Converting rectangular equations to polar equations is quite simple. Recall that x = r cos θ and y = r sin θ. By using this information, the polar equation that corresponds with the rectangular equation of 3 x – y + 2 = 0 can be found.

Example:

Example:

Convert Rectangular Equations to Polar Equations Example 2: Convert y = 3 to polar form. Since y = r sin , substitute into the equation. r sin = 3 Solve for r when possible. r = 3 / sin r = 3 csc is the equivalent polar equation.

Convert Rectangular Equations to Polar Equations Example 3: Convert (x - 3)2 + (y + 3)2 = 18 to polar form. Square each binomial. x 2 – 6 x + 9 + y 2 + 6 y + 9 = 18 Since x 2 + y 2 = r 2, re-write and simplify by combining like terms. x 2 + y 2 – 6 x + 6 y = 0 Substitute r 2 for x 2 + y 2, r cos for x and r sin for y. r 2 – 6 rcos + 6 rsin = 0 Factor r as a common factor. r(r – 6 cos + 6 sin ) = 0 r = 0 or r – 6 cos + 6 sin = 0 Solve for r: r = 0 or r = 6 cos – 6 sin

Convert Polar Equations to Rectangular Equations The goal is to change all r’s and ’s to x’s and y’s. Example 1: Convert r = 5 to rectangular form. Since r 2 = x 2 + y 2, square both sides to get r 2 = 25 Substitute. x 2 + y 2 = 25 is the equivalent polar equation to r = 5

Convert Polar Equations to Rectangular Equations Example 2: Convert r = 5 cos to rectangular form. Multiply both sides by r r 2 = 5 r cos Substitute x for r cos r 2 = 5 x Substitute for r 2. x 2 + y 2 = 5 x is rectangular form.

Convert Polar Equations to Rectangular Equations Example 3: Convert r = 2 csc to rectangular form. Since csc ß = 1/sin , substitute for csc . Multiply both sides by 1/sin. Simplify y = 2 is rectangular form.

Polar Graphs You will notice that polar equations have graphs like the following:

Relations between Cartesian, Cylindrical, and Spherical Coordinates