CO1301 Games Concepts Week 18 Basic Trigonometry

1
CO1301 - Games ConceptsWeek 18Basic
Trigonometry

• Gareth Bellaby

2
References

• Rabin, Introduction to Game Development, Chapter
  4.1
• Van Verth Bishop, Essential Mathematics for
  Games, Appendix A and Chapter 1.
• Eric Lengyel, Mathematics for 3D Game Programming
  Computer Graphics
• Frank Luna, Introduction to 3D Game Programming
  with Direct 9.0c A Shader Approach, Chapter.1.

3
Lecture Structure

• Introduction
• Trigonometric functions sine, cosine, tangent
• Circles
• Useful trigonometric laws

4
Why do Trigonometry?

• Why is trigonometry relevant to your course?
• Games involve lots of geometrical calculations
• Rotation of models.
• Line of sight calculations.
• Collision detection.
• Lighting.
• For example, the intensity of directed light
  changes according to the angle at which it
  strikes a surface.
• You require a working knowledge of geometry.

5
Introduction

• Mathematical Functions
• A mathematical function defines relationship
  between one variable and another. A function
  takes an input and calculates an output
  according to some rule or operation. For
  instance, sine function takes an angle as input
  and calculates a number.
• Mathematical Laws
• I'll introduce some laws. I'm not going to prove
  or derive them. I'm going to ask you to accept as
  being true.

6
Greek letters

• It is a convention to use Greek letters to
  represent angles and some other mathematical
  terms.

7

• Part 2 Trigonometric functions sine, cosine,
  tangent

8
Trigonometry

• Trigonometry arises out of an observation about
  right angled triangles.
• Take a right angled triangle and consider one of
  its angles (not the right angle itself). We'll
  call this angle a.

• The opposite side to the angle is y.
• The nearest side to the angle is x.
• The longest side of the triangle is h.

9
Trigonometry

• There is a relationship between the angle and the
  lengths of the sides. This relationship is
  expressed through a trigonometric function, e.g.
  sine (abbreviated to sin).

sin(a) y / h
10
Values of sine
11
Trigonometry
You need to be aware of three trigonometric
functions sine, cosine and tangent.
12
Trigonometry
13
Trigonometric Functions

• Sine, cosine and tangent are mathematical
  functions.
• There are other trigonometric functions, but they
  are rarely used in computer programming.
• Angles can be greater than 2p or less than -2p.
  Simply continue the rotation around the circle.
• You can draw a graph of the functions. The x-axis
  is the angle and the y-axis is (for example)
  sin(x). If you graph out the sine function then
  you create a sine wave.

14
Sine Wave and Cosine Wave
Image taken from Wikipedia
15
Tangent Wave
Image taken from Wikipedia
16
C
include "math.h" OR USE include
ltcomplexgt using namespace std

• C has functions for sine, cosine and tangent
  wiithin its libraries. Use the maths or complex
  libraries
• The standard C functions use radians, not
  degrees.

float rad float result result
sin(rad) result cos(rad) result tan(rad)
17
Radians

• A degree is a fairly arbitrary measurement of
  rotation. More common to use radians.
• A complete revolution is equal to 2p radians.
• 2p radians is equal to 360 degrees.

18
PI

• Written using the Greek letter p.
• Otherwise use the English transliteration "Pi".
• p is a mathematical constant.
• p is the ratio of the circumference of a circle
  to its diameter. This value holds true for any
  circle, no matter what its size. It is therefore
  a constant.
• 3.14159 (approximately).

19

• Part 3 Circles

20
Circles

• The constant p is derived from circles so useful
  to look at these.
• A basic shape.
• Circumference is the length around the circle.
• Diameter is the width of a circle at its largest
  extent, i.e. the diameter must go through the
  centre of the circle.
• Radius is a line from the centre of the circle to
  the edge (in any direction).

21
Circles

• Tangent is a line drawn perpendicular to the end
  point of a radius. You'll know these from drawing
  splines (curves) in 3ds Max. You'll see them
  again when you generate splines in graphics and
  AI.
• Chord is line connecting two points on a circle.

22
Circles

• Segment is that part of a circle made by chord,
  i.e. a line connecting two points on a circle.
• Sector is part of a circle in which the two edges
  are radii.

23
Circle

• Using Cartesian coordinates.
• Centre of the circle is at (a, b).
• The length of the radius is r.
• The length of the diameter is d.

24
Points on a Circle

• Imagine a line from the centre of the circle to
  (x,y)
• a is the angle between this line and the x-axis.

25

• Part 4 Useful trigonometric laws

26
Identities
27
Trigonometric Relationships

• This relationship is for right-angled triangles
  only.

Where
28
Trigonometric Relationships

• These relationships are for right-angled
  triangles only.

29
Properties of triangles

• This property holds for all triangles and not
  just right-angled ones.
• The angles in a triangle can be related to the
  sides of a triangle.

30
Properties of triangles

• These hold for all triangles

31
Inverses

• Another bit of terminology and convention you
  need to be familiar with.
• An inverse function is a function which is in the
  opposite direction. An inverse trigonometric
  function reverses the original trigonometric
  function, so that
• x sin(y) y arcsin(x)
• The inverse trigonometric functions are all
  prefixed with the term "arc" arcsine, arccosine
  and arctangent.
• In C asin() acos() atan()

32
Inverses

• The notation sin-1, cos-1 and tan-1 is common.
• We know that trigonometric functions can produce
  the same result with different input values, e.g.
  sin(75o) and sin(105o) are both 0.97.
• Therefore an inverse trigonometric function
  typically has a restricted range so only one
  value can be generated.

33
Inverses
Function
Domain
Range