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UFCEKU-20-3Web Games Programming Game Math

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UFCEKU-20-3Web Games Programming Agenda Revise some basic concepts Apply Concepts to Game Elements

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UFCEKU-20-3Web Games Programming The Natural Numbers 1 2 3 4 5 6 7 The Natural Numbers 1..infinity

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UFCEKU-20-3Web Games Programming The Integers -3 -2 0 1 2 3 The integers = whole numbers positive, negative or zero Computer data type int, uint (AS3) name depends on language in use

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UFCEKU-20-3Web Games Programming Rational Numbers -3 -2 0 1 2 3 The Rationals = fractions with an exact decimal value e.g 1/2 =.5 positive or negative Computer data type long, double,float, real, Number (AS3) name depends on language in use

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UFCEKU-20-3Web Games Programming Irrational Numbers -3 -2 0 1 2 3 The irrationals = fractions without an exact decimal value e.g 1/3 =.33333 positive or negative Computer data type as for rationals long, double,float, real, Number

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UFCEKU-20-3Web Games Programming The Number Line: Root 2 0 1 2 3 4 5 Square root of 2 1.41421

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UFCEKU-20-3Web Games Programming The Golden Ratio - Phi 0 1 2 3 4 5 The Golden Ratio 1.618033897

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UFCEKU-20-3Web Games Programming The Natural Logarithmic Constant e 0 1 2 3 4 5 e = 2.7182818284…

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UFCEKU-20-3Web Games Programming The Number Line 0 1 2 3 4 5 PI the Relationship between the diameter of a circle and its circumference 3.142 (easy) 3.141592654 (calculator) valueForPI = Math.PI; // AS3 (3.141592653589793) Hmm… pie

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UFCEKU-20-3Web Games Programming The Number Line Attributes of a Circle: Radius Diameter Circumference Area Radius =r Diameter =2r Circumference = C = PI x D Area = PI x r x r

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UFCEKU-20-3Web Games Programming ActionScript const PI:Number = Math.PI;// define a constant public function areaOfCircle(radius:Number):Number { var area:Number; area = PI * radius * radius; return area; }

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UFCEKU-20-3Web Games Programming Radians Radian is a natural measurement of angular rotation There are 2PI radians in a circle (~= 6.284)

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UFCEKU-20-3Web Games Programming Radians v Degrees There are 2 PI radians in a circle = 3.142 x 2 = 6.284 There 360 degrees in a circle One radian = 360 / 6.284 approximately 57.288 degrees Formula for changing radians to degrees : degrees = (360 / 2PI) Formula for changing degrees to radians : radians = degrees x (2PI / 360)

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UFCEKU-20-3Web Games Programming Pythagoras a b c a 2 = b 2 + c 2

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UFCEKU-20-3Web Games Programming Pythagoras a2a2 b2b2 c2c2

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UFCEKU-20-3Web Games Programming Pythagoras a = 5 c = 3 b = 4

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UFCEKU-20-3Web Games Programming Pythagoras a 2 = 4 2 + 3 2 a = 16 + 9 a = SQRT (25) a = 5

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UFCEKU-20-3Web Games Programming Application to Games x y P1 (x1, y1) P2 (x2, y2) Length = x2 - x1 Length = y2 - y1

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UFCEKU-20-3Web Games Programming Application to Games x y P1 (10, 15) P2 (80,50 ) Length = x2 - x1 = 80 - 10 = 70 Length = y2 - y1 = 50-15 = 35 a

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UFCEKU-20-3Web Games Programming Application to Games x y a distance a = SQRT((x2-x1) 2 + (y2 - y1) 2 )

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UFCEKU-20-3Web Games Programming ActionScript // distanceXY calculates the distance between two points x and y // and returns the resulting value public function distanceXY(pointX1, pointY1, pointX2, pointY2:int):Number{ var distance:Number; distance = Math.sqrt((pointX2-pointX1) * (pointX2-pointX1) + (pointY2-pointY1) * (pointY2-pointY1)); return distance }

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UFCEKU-20-3Web Games Programming Trigonometry 1 1 cosine sInesIne Tan

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UFCEKU-20-3Web Games Programming Sine Values 1 1 cosine sInesIne Rotate line through some angle Read value off sine axis

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UFCEKU-20-3Web Games Programming 33 degrees Sine read sine value angle about 0.52

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UFCEKU-20-3Web Games Programming Cosine Values 1 1 cosine sInesIne angle 45 degrees

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UFCEKU-20-3Web Games Programming Tan Values 1 1 tan Rotate line through some angle (45) Read value off tan axis tan tan(45) = 1

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UFCEKU-20-3Web Games Programming Tan Values 1 1 cosine sInesIne Rotate line through some angle (65 degrees) Read value off tan axis (about 2.7)

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UFCEKU-20-3Web Games Programming Tan Values 1 1 cosine sInesIne Rotate line through some angle (89 degrees) Read value off tan axis (about 57.289)

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UFCEKU-20-3Web Games Programming Tan 90 ! 1 1 cosine sInesIne Rotate line through some angle (90 degrees) Lines parallel - tan 90 not defined

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UFCEKU-20-3Web Games Programming Tan > 90 1 1 cosine sInesIne Rotate line through some angle (135 degrees) value read from this axis (-1)

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UFCEKU-20-3Web Games Programming Trigonometric Ratios sine of the angle = opposite / hypotenuse cosine of the angle = adjacent / hypotenuse tan of the angle = opposite / adjacent hypotenuse adjacent Angle in degrees = 1/sine etc. opposite

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UFCEKU-20-3Web Games Programming Trig Ratios sine of the angle = opposite / hypotenuse cosine of the angle = adjacent / hypotenuse tan of the angle = opposite / adjacent SOH CAH TOA

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UFCEKU-20-3Web Games Programming Finding the angle to a point point(x1,y1) Point (x2, y2) angle theta?

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UFCEKU-20-3Web Games Programming Finding the angle to a point point(x1,y1) Point (x2, y2)

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UFCEKU-20-3Web Games Programming Finding the angle to a point point(x1,y1) Point (x2, y2) angle theta?

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UFCEKU-20-3Web Games Programming Use the tangent formula point(x1,y1) point (x2, y2) angle theta tan theta = opposite / adjacent tan theta= (y2-y1) / (x2-x1) the angle = 1/tan (theta) atan(theta)

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UFCEKU-20-3Web Games Programming Calculate for each frame point(x1,y1) point (x2, y2) (moving sprite) angle theta?

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UFCEKU-20-3Web Games Programming Some tan angles point to the same value… 1 1 cosine sInesIne tan 225 =1 tan 45 =1

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UFCEKU-20-3Web Games Programming Using the atan2 function arctan(x) function may return same angle value for two different points if in certain quadrants - would need to determine which is correct for direction required Use atan2(y,x) specially designed function, which by using numbers in the complex plane, returns one unique value for any angle Value is returned in radians - convert to degrees with appropriate formula

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UFCEKU-20-3Web Games Programming ActionScript // definition in GameMath.as public function pointAtSprite(xPoint:Number, yPoint:Number, pointerX:Number, pointerY:Number):Number { var dx:Number = xPoint - pointerX; var dy:Number = yPoint - pointerY; // determine angle and convert to degrees var pointAngle:Number = Math.atan2(dy,dx); var pointDegrees:Number = 360*(pointAngle/(2*Math.PI)); return pointDegrees; } //pointAtSprite stage.addEventListener(Event.ENTER_FRAME, getPointAngle); function getPointAngle(event:Event){ pointAtSpriteAngle= gameMath.pointAtSprite(boat_mc.x, boat_mc.y, pointer.x, pointer.y); pointer.rotation = pointAtSpriteAngle;// instance name of gun turret sprite };

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UFCEKU-20-3Web Games Programming Game Math:Applications Used to calculate sprite positions and directions Determine distances between game objects (Pythagoras) Use distance between objects for collision detection purposes Change game state based on game sprites positions in game world Give sprites speed and a known direction (Trigonometry SOH CAH TOA ) Use atan2 to follow or point at moving sprite In 3D games, math is used to orientate objects in three coordinate systems x,y,z Used to construct formulae which implements sprite behaviours for physics - such as acceleration, momentum, friction.

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UFCEKU-20-3Web Games Programming The Prime Numbers 2 35 7 The distribution of Prime numbers is the most important unanswered question in Mathematics - can you find a formula to say if a given number is a Prime number? 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 If you can you will probably win a Nobel prize and be famous forever! A Prime number is only divisible by 1 and itself

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4.3 Trigonometry Extended: The Circular Functions

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