1. 3D Game Engine Design 1 3D Game Engine Design Ch. 2.5. 3D MAP LAB

2. 3D Game Engine Design 2 2.5. Standard 3D Objects  Useful as bounding regions for  Rapid culling in the rendering process  Rapid determining of the collision detection of two objects  Primitives  Spheres  Oriented Boxes  Capsules, Lozenges  Ellipsoids and etc.

3. 3D Game Engine Design 3  Center Point and Radius with points  Sphere Containing Axis-Aligned Box  Compute the minimum-volume axis- aligned bounding box  Select the smallest enclosing sphere of the box with centered at the box center  Rapid, but not as good a fit. Spheres

4. 3D Game Engine Design 4 Spheres Spheres (cont’d)  Sphere Centered at Average of Points  Sphere Center : Average of Points  Sphere Radius : Smallest value from center to points enclosing all points  2x Iteration than Axis-Aligned box method  Minimum-Volume Sphere (Welzl 91)  Uses a randomized linear algorithm  Maintain a set of supporting points( lying on the sphere) while processing the input point set at a time.

5. 3D Game Engine Design 5 Oriented Boxes  Oriented Box  Provide a better fit than spheres  Defined by  A center  3 Orthonormal Axes for  3 extents of Axes , points inside or on the box  Orthonormal matrix  where

6. 3D Game Engine Design 6 Oriented Boxes Oriented Boxes (cont’d)  Axis-Aligned Box  Find the extreme points and  Fitting with a Gaussian Distribution  Gaussian Distribution  : mean of the distribution  : the covariance matrix of the distribution  Fitting points with anisotropic gaussian distribution

7. 3D Game Engine Design 7 Oriented Boxes Oriented Boxes (cont’d)  Fitting with a Gaussian Distribution  Axes of the box : unit-length eigenvectors of covariance matrix M  Extents :

8. 3D Game Engine Design 8 Oriented Boxes Oriented Boxes (cont’d)  Minimum-Volume Box  Best-fitting box : the minimum volume containing the points  Iterative scheme for a minimization  For any coordinate axes, points are projected onto the axes  The values :

9. 3D Game Engine Design 9 Oriented Boxes Oriented Boxes (cont’d)  Iterative Minimization for M-V Box  Center of the minimum-volume box  Extents of the minimum-volume box  Selection of coordinate axes  : unit length vector and an angle  The volume of Box :  Find that minimizes the volume

10. 3D Game Engine Design 10 Oriented Boxes Oriented Boxes (cont’d)  Fitting Triangles with Gaussian Distribution  Data points : vertices of a triangle mesh  Fit of an oriented box to the convex hull of the vertices  triangles, i-th triangle  Triangle and its interior  Mean point of the convex hull

11. 3D Game Engine Design 11 Oriented Boxes Oriented Boxes (cont’d)  Fitting Triangles with Gaussian Distribution  Covariance matrix of the convex hull  Axes of Oriented Box : unit length eigenvectors of M  Extents of Oriented Box :

12. 3D Game Engine Design 12Capsules  Capsule  The set of all points that are distant from a line segment with end point and direction  Shape : a cylinder that has 2 hemispherical caps at the end points  Least square fitting  Line fitting :, where average of the data points and unit-length direction vector  to be the maximum distance from the data points to Line

13. 3D Game Engine Design 13 Capsules Capsules (cont’d)  Least Square Fitting  Select unit axes vectors and set Data points  Find the domain of for the 2 hemisphere at end  Find the largest so that all points lie above the hemisphere

14. 3D Game Engine Design 14 Capsules Capsules (cont’d)  Least Square Fitting  Find the smallest so that all points lie below the hemisphere  End points :  Minimum of minimum-Area Projected Circles  Projection of points onto the Plane  Compute the minimum-area circle of projected points and find and

15. 3D Game Engine Design 15 Lozenges  Lozenges  Set of all points that are distant from a rectangle with origin and edge directions  Shape : an oriented rectangle with  Attached 4 half-cylinder sides  4 quarter-spherical corners

16. 3D Game Engine Design 16 Lozenges  Fitting with a Gaussian Distribution  Compute the mean of the points and the covariance matrix M  Unit-length eigenvectors of the M :  Data points :  Data points are bounded by 2 planes  Lozenge radius :

17. 3D Game Engine Design 17 Lozenges Lozenges (cont’d)  Fitting with a Gaussian Distribution  Lozenge edge : end of the hemicylinder  Find the Lozenge edge  Analogous to the fitting by 3D capsule  2D capsule containing  The largest so that all points lie above the hemicircle

18. 3D Game Engine Design 18 Lozenges Lozenges (cont’d)  Fitting with a Gaussian Distribution  Find the Lozenge edge  The smallest so that all points lie below the hemicircle  The Lozenge edge

19. 3D Game Engine Design 19 Cylinders  Cylinder  A distance from a line  Least Squares Line Contains Axis  Fit the points by a line using the least- square algorithm , where average of the data points and unit-length direction vector  Select unit axes vectors and set Data points

20. 3D Game Engine Design 20 Cylinders Cylinders (cont’d)  Least Squares Line Contains Axis  Cylinder radius :  Cylinder height :  Modified Axis  Least Squares Line Moved to a Minimum-Area Center  Minimum-area circle containing is computed and has center

21. 3D Game Engine Design 21 Ellipsoids  Ellipsoid  Center  Orthonormal axis directions  The ellipsoid with center and axes

22. 3D Game Engine Design 22 Ellipsoids Ellipsoids (cont’d)  Axis-Aligned Ellipsoid  Firstly generate the axis-aligned box containing the points  be the vectors of the min/max box components  Compute the ratios of semi-axis lengths  Semi-Axis Lengths

23. 3D Game Engine Design 23 Ellipsoids Ellipsoids (cont’d)  Fitting Points with a Gaussian Distribution  Similar to the case of an oriented box  Center of the ellipsoid : the mean of points  Axes : eigenvectors of the covariance matrix  Semi-Axis Lengths : eignevalues

24. 3D Game Engine Design 24 Ellipsoids Ellipsoids (cont’d)  Minimum-Volume Ellipsoid  Random Linear Techniques (Welzl 91)  Constrained numerical minimization  Difficult to implement  Not possible for real-time apps.