CO1301: Games Concepts Dr Nick Mitchell (Room CM 226) Material originally prepared by Gareth Bellaby Lecture 5 Using Vectors for Collision Detection

Reading about Vectors  Rabin, Introduction to Game Development:  4.2: "Collision Detection and Resolution"  Van Verthe, Essential Mathematics for Games:  14.6: "A Simple Collision System"

Lecture Outline Vector practice (Revision) Collision detection and resolution

Topic 1:A reminder about Vectors

Right-angled triangles  Remember that with a right-angled triangle the length of the longest side (called the hypotenuse) is equal to the square root of the sum of the squares of two shorter sides. (Pythagoras’ Theorem)

Vector between two points  The vector V going from point P1 to point P2 is P1 subtracted from P2 (i.e. P2 – P1)  Remember that P1, P2 and V are all expressed as Cartesian co-ordinates.  The vector V from P1 to P2 is: = V( X 2 - X 1, Y 2 - Y 1, Z 2 - Z 1 )

Length of a vector  A vector makes a right-angled triangle in 3 dimensions... ...so use Pythagoras’ Theorem about right-angle triangles to calculate the length of the vector.  If we have a vector V(X, Y, Z) then:

Normalised vector  A normal is any vector whose length is 1.  It is also caled the direction vector.  A vector is converted to a normal (i.e. normalised) by dividing the components by the length:

Topic 2:Collision Detection

Collision Detection  Collision detection is a common operation.  You often need to able to determine whether two objects in a game have come into contact:  A bullet hitting a target  A racing car reaching the winning point  The player-character trying to move through a wall  Checking whether an event has triggered, e.g. a triggered piece of dialogue or a monster jumping out.  Distance calculations also come under this topic.

Point to Point Collsions  The objects move frame by frame. Calculate the distance between the two points each frame.

Point to Point  Point to point is the simplest case.  Example: a bullet striking a bull's eye.  Point to point is simply a matter of working out the distance between two points. This is the same calculation as the length of a vector.

Bounding volumes  Most objects in a game are not points so you cannot use point to point calculations.  It is time consuming and mathematically complex to test whether two complex objects intersect.  Instead use a simplified shape that fits around the object – a bounding volume.

Bounding volumes  Complex objects:  Bounding spheres:

Bounding volumes  Make the bounding volume as close a fit as possible.  Spheres and boxes are two commonly used shapes.  The technique sacrifices precision for efficiency (speed) – an issue which crops up frequently in games programming...  Bounding volumes are also used for a initial coarse check before engaging in more fine detailed analysis,  e.g. a bounding sphere can be employed to see whether there is any chance of collision.

Point to Sphere collisions  Check to see whether a point has collided with a sphere.  The radius of a sphere is the distance from the centre of the sphere to the edge of the sphere.  Calculate the distance from the point to the centre of the sphere.  If the distance is smaller than the radius of the sphere then a collision has occurred.

Point to Sphere

Sphere to Sphere Collisions  Surround both object with close fitting spheres.  The radius of a sphere is the distance from the centre of the sphere to its edge.  Calculate the distance from the centre of the first sphere to the centre of the second sphere.  If the distance is smaller than the sum of the sphere radii then a collision has occurred.

Sphere to Sphere

More bounding volumes  Spheres are useful because the maths is comparatively simple - it's just an extension of vector calculations.  Spheres also suit some objects, e.g. people (you would probably use a cylinder in practice... ...but 2 dimensionally, a sphere (i.e. a circle) is equivalent to a cylinder (with zero height).  But spheres do not suit all objects, e.g. a crate or a house.  In these cases a box (cuboid) would be a more appropriate approximation.

Bounding boxes  The simplest box is an "axis-aligned bounding box”.  The box is exactly aligned with the axes, e.g. horizontally and vertically.

Bounding boxes  A box which is not axis-aligned will be more difficult to use.  However, some objects fit better inside a box than a sphere.  A problem arises if the object can rotate. Boxes better suit objects that cannot rotate.  Of course, it is possible to do the calculations for a non-axis aligned bounding box. However, I am going to leave this until we move on to planes and point to plane collisions...

Point to Axis-Aligned Box  Check the position of the point against the minimum and maximum x, y and z values of the box.

Sphere to Box Collision  The easiest way to implement this is to do a bit of sideways thinking...  The sphere collides with the box when its edge crosses over the edge of the box.  This can only happen if the distance from the centre of the sphere to the edge of the box is smaller than the radius of the sphere.  So lets expand the size of the bounding box by the radius of the sphere.  Now do a simple point to box calculation instead.

Sphere to Box  Same method as point to box, but add the radius to the maximum x, y and z values of the box, and subtract it from the minimum x, y and z values. Note that this method is inaccurate at the corners...

Collision Resolution  The order of operation is always:  Collision Detection  Collision Resolution  Need to have a resolution because:  An object moves over the course of one frame to the next  If an object collides with an obstacle and we do not attempt to resolve the collision immediately...  The object could become stuck because it may well be inside the obstacle.

Collision Resolution  Moving frame- by-frame...  Therefore the object is inside the obstacle when the collision is detected...  Hence it will become stuck inside the object.

Simple Collision Resolution Algorithm  Move the model.  Record the movement vector you used to move the model (x, y, z).  Check if a collision has occurred.  If a collision has occurred:  Move the model back to where it was at the beginning of the frame (-x, -y, -z)  Note that this is simplistic.  You should check to see where along the path of movement the collision occurred and set the position to this location  You will look at the maths to do it at a later time.