1. Some Cool Tricks

2.  We can consider the screen as high school graph paper.  Each sprite or object is located somewhere in the coordinate system.

3.  Remember our screen is not like the school mathematical graph paper! The origin is at the top-left!  That means the bottom of our object is at a higher y position than the top!

4.  Moving things around is called translation.  Usually we translate our object to the origin before we perform rotation and scaling.

5.  Move every point in one object to another location.

6.  For rotation, lets visit the Siggraph web page: https://www.siggraph.org/education/materials/HyperGraph/modeling/mod_tran/2drota.htm, accessed August 2014. https://www.siggraph.org/education/materials/HyperGraph/modeling/mod_tran/2drota.htm

7.  We use Pythagoras to find the distance between two points.

8.  Using (x, y) positions is common.  However, you may use (r, Θ)  r = distance from origin  Θ = angle.  Converting between the two systems is easy.  Just remember the name!

9.  Beware, we need a little bit more sophistication to calculate the angle… as this gives us ambiguity.

10.  We can use circle to circle collision as an example.  We find the centre of each circle and use Pythagoras to discover the distance between the two.  The equations to the right tell us when there has been a collicsion! We can possibly do this with square distances to avoid using expensive sqrt operations. We must be careful to avoid interpenetration as the entities will indefinite collide with each other.

11.  Interpenetration of objects.  Interlocking of objects.  Quantum Tunnelling Effects.

12.  Once again in scaling we need to translate our object’s origin to the origin of the world, we then perform the scaling operation.  Remember to translate back!

13.  We use logic all the time in coding.  We will use logic for our AI and decision making in games.  We will explicitly cover some rudimentary logic.

14.  And  Or  Not  Xor  Implies  Tautologies—Saying the same thing more than once.  Associative Laws—Order does not matter.  Distributive Laws—We can say things in different ways that mean the same thing.  Identity Laws

17.  Sets can be used to make decisions.  Sets indicate belongingness.

18.  Venn Diagrams  Union  Intersection  Difference.  Sets  Subsets  Compliment  Universal Set  Identity Laws

20.  What is in A, B and C?  What is in A?  What is in not A?  Useful for games.  Useful for determining strategies.

21.  Yes!  std::set!  How awesome is that?

22.  Vectors and matrices are mathematically convenient tools.  They are used in games for graphical mathematics.  So annoying that std::vector is named in such a way to make things a little confusing.

23. I like to call this Quantum Tunnelling. We miss the detection of a collision as the bullet passes through the wall.

24. Bounding box collision detection in 2D.

25.  Tricks  One’s Compliment

26.  The distance between two points can be calculated using Pythagoras.  This can be used for simple collision detection.  Movie, https://www.youtube.com/watch?v=pVo6szY E13Y, accessed August 2014. https://www.youtube.com/watch?v=pVo6szY E13Y

27.  https://www.youtube.com/watch?v=o- OvVep2uSg, accessed August 2014. https://www.youtube.com/watch?v=o- OvVep2uSg

28.  Analytical Geometry is concerned with points and the relationships between points.  We can find angles between two points to determine interesting relationships.

29.  Useful for determining:  (a) If a collision would take place.  (b) If a collision has taken place.

30.  Let’s look at the diagrams!  http://www.intmath.com/plane-analytic- geometry/perpendicular-distance-point- line.php, accessed August 2014. http://www.intmath.com/plane-analytic- geometry/perpendicular-distance-point- line.php

31.  Vectors can represent acceleration, velocity and displacement and any sort of force, which is essential to any physics orientated game.  It allows for you to calculate paths, trajectories and many aspects of physics within code.

32.  A vector is a point.  For two dimensional mathematics:  (x, y).  These vectors have interesting operations.  They make it easier to do mathematics.

33.  We can simply calculate the resulting vector when combining multiple forces…

34.  A matrix is a column by row matrix of values.  The matrix has defined operations so we can perform mathematical operations efficiently on a collection of points.

38.  Read the wikipage on 2D Graphics.  This will help you understand more about your programming project.  http://en.wikipedia.org/wiki/2D_computer_graphics http://en.wikipedia.org/wiki/2D_computer_graphics