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Geometry Primer Lines and rays Planes Spheres Frustums Triangles Polygon Polyhedron.

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Presentation on theme: "Geometry Primer Lines and rays Planes Spheres Frustums Triangles Polygon Polyhedron."— Presentation transcript:

1 Geometry Primer Lines and rays Planes Spheres Frustums Triangles Polygon Polyhedron

2 Lines and Rays Given 2 points, we can parametrically define a line which passes through these two points as  P(t) = (1-t)P 1 + tP 2 The line segment between these two points corresponds to all values of t between zero and one A ray is defined parametrically as  P(t) = P o + tV Where t is greater than or equal to zero  Allowing negative values of t will give a valid equation for a line

3 Planes Plane equation  Ax + By + Cz + D = 0 A, B, and C are the respective x, y, and z coordinates of a vector normal to the plane D = -N.P o  N is the plane normal vector  P o is an arbitrary point on the plane If N is a unit vector, the equation  d = N.Q + D Gives the distance from the plane to an arbitrary point Q If d > 0, Q lies on the positive side of the plane. Otherwise Q lies on the negative side of the plane.  Use this to test for path-plane collision

4 A Different Plane Equation Point – normal form  N.(P – P o ) = 0 P and P o are arbitrary points on the plane  Stored as a point and a normal in many applications, or calculated on the fly from three points. Given by equation  N.Q + D = 0  Line – Plane intersection Example

5 Spheres Sphere equation  (x - x 0 ) 2 + (y - y 0 ) 2 + (z - z 0 ) 2 = r 2  r is the radius, and is centered at (x 0,y 0,z 0 ) In an application, store the center point and the radius.

6 Frustums A frustum as a volume of space containing everything visible in a scene. A frustum can be defined by a:  Near plane (or near plane distance)  Far plane (or far plane distance)  Field of view FOV x or FOV y ?  Aspect ratio Alternatively, it can be described by the six planes which bound it.

7 Frustum Illus.

8 Triangle

9 Polygon

10 Polyhedron Convex polyhedrons are also known as polytopes, and are often used to approximate non-convex meshes where speed is an issue. Also known as brushes.

11 Applied Geometry Collision models for games can be approximated using fundamental shapes  E.g. Use boxes for limbs of characters, 3 spheres for an ant…

12 Visibility Determination *Bounding box test Bounding sphere test

13 Bounding Box Axis aligned, or AABB  Find minimum and maximum points and use these to construct box Oriented, or OBB  Find natural axes for points  Compute dot product of each vertex position with unit vectors R, S, T of the natural axis and take the minimum and maximum values Planes are given by:

14 Collision Detection Sphere – Sphere Sphere - Plane *AABB – AABB *OBB – OBB *Box – Plane Ray – Plane Ray – Triangle

15 Ray – Triangle Intersection Example Investigate other methods  Badoul’s algorithm  Moller-Haines’ algorithm  D3DX algorithm D3DX common source

16 Usage / Examples Detecting when a user has clicked an object Determining if, when, and where objects have collided  Decide how to respond to collision

17 Reading Chapters 4, 5, 7, 8 in Mathematics for 3D Game Programming and Computer Graphics cover the material presented here

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