CO1301: Games Concepts

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CO1301 Games Concepts
Lecture 8 Basic Trigonometry

• Dr Nick Mitchell (Room CM 226)
• email npmitchell_at_uclan.ac.uk
• Material originally prepared by Gareth Bellaby

Hipparchos the father of trigonometry (image
from Wikipedia)
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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

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Lecture Structure

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

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Why study 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.

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Mathematical Functions

• A mathematical function defines a relationship
  between one variable and another.
• A function takes an input (argument) and relates
  it to an output (value) according to some rule or
  formula.
• For instance, the sine function maps an angle (as
  input) onto a number (as output).
• The set of possible input values is the functions
  domain.
• The set of possible output values is the
  functions range.
• For any given input, there is exactly one output
• The 32 cannot be 9 today and 8 tomorrow!
• Mathematical Laws
• I'll introduce some laws. I'm not going to prove
  or derive them. I will ask you to accept them as
  being true.

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Greek letters

• It is a convention to use Greek letters to
  represent angles and some other mathematical
  terms
• a alpha
• ß beta
• ? gamma
• ? theta
• ? lambda
• p pi
• ? (capital) Delta

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Trigonometry

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

o
a
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Trigonometry

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

sin(a) o / h
o
a
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Values of sine
degrees sin (degrees)
0 0
15 0.26
30 0.5
45 0.71
60 0.87
75 0.97
90 1
105 0.97
120 0.87
135 0.71
150 0.5
165 0.26
degrees sin (degrees)
180 0
195 -0.26
210 -0.5
225 -0.71
240 -0.87
255 -0.97
270 -1
285 -0.97
300 -0.87
315 -0.71
330 -0.5
345 -0.26
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Trigonometry
You need to be aware of three trigonometric
functions sine, cosine and tangent.
Function Name Symbol Definition
sine sin sin(a) o / h
cosine cos cos(a) a / h
tangent tan tan(a) o / a sin(a) / cos(a)
o
a
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Radians

• You will often come across angles measured in
  radians (rad), instead of degrees (deg)...
• A radian is the angle formedby measuring one
  radiuslength along the circumferenceof a
  circle.
• There are 2p radians in acomplete circle (
  360)
• deg rad 180 / p
• rad deg p / 180

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Trigonometry
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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.

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Sine Wave and Cosine Wave
Image taken from Wikipedia
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Tangent Wave
Image taken from Wikipedia
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C

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

include ltcmathgt using namespace std
float rad float result result
sin(rad) result cos(rad) result tan(rad)
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PI

• Written using the Greek letter p.
• Otherwise use the English transliteration "Pi".
• p is a mathematical constant.
• 3.14159 (approximately).
• 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.

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Circles

• The constant p is derived from circles so useful
  to look at these.
• Circles are 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).

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Circles

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

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Circles

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

sector
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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.

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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.

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Identities
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Trigonometric Relationships

• This relationship is for right-angled triangles
  only

Where
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Trigonometric Relationships

• These relationships are for right-angled
  triangles only

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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.

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Properties of triangles

• These hold for all triangles

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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
• If x sin(y) then y arcsin(x)
• The inverse trigonometric functions are all
  prefixed with the term "arc" arcsine, arccosine
  and arctangent.
• In C asin() acos() atan()

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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.

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Inverses