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CO1301 Games Concepts Week 18 Basic Trigonometry

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This relationship is expressed through a trigonometric function, e.g. sine (abbreviated to sin) ... Sine, cosine and tangent are mathematical functions. ... – PowerPoint PPT presentation

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Title: 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
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