Copyright © Curt Hill Visualization of 3D Worlds How are these images described?
Copyright © Curt Hill Representation There are many ways to represent the data –Collections of planar polygon –Polyhedrons –Formulas –Constructive Solid Geometry –Deformation –Fractals –Quadtrees
Copyright © Curt Hill Polygons Usually triangles, which are always planar Describe the three vertices –Each vertex is a triple of x, y, z value Rectangles may always be constructed from two triangles –Four arbitrary points do not have to be planar –Any other polygon may also be so constructed
Copyright © Curt Hill Polyhedrons Solids may be constructed from polygons if each vertex point is a vertex of several polygons We may also create solids with simple formulae –A sphere as a simple formula –Specify an origin and radius –Other polyhedrons may also be constructed in this way
Minecraft Most things in Minecraft are variation of blocks These are placed in an x, y, z coordinate system This gives the block look of the game The ease of use allows creative mode Copyright © Curt Hill
Sweeping Volumes Take a 2D figure and rotate on an axis A rectangle rotated on any of its sides produces a cylinder A rectangle rotated on a diagonal produces something like a double cone A circle rotated around a line that does not intersect it produces a toroid
Copyright © Curt Hill Dragging Volumes Take a 2D figure and pull it through space on a known trajectory A pipe is a circle drug in a straight line A hose is a circle drug in a curve Sometimes known as extrusion Since most rooms and buildings are rectanguloid solids dragging a rectangle on a straight line works well
Copyright © Curt Hill Ruled Surfaces Sweeping out a surface with a line that can move Line is called a directrix If the lines are parallel we may get a fence or paper roll shape If the lines are not parallel can get variance in three dimensions
Copyright © Curt Hill More Ruled Surfaces Cylinder is when one point of directrix stays in a circle in a plane and directrix is normal to plane A cone is made with a directrix moved through a point An irregular cone-like item could also result if the directrix does not trace a circle
Copyright © Curt Hill Mesh Surfaces Take a series of 3D point coordinates from a surface Subdivide the surface into triangles Each point is the vertex of several triangles This mesh may approximate any surface
Copyright © Curt Hill Blobby objects Most of the previous items had relatively simple mathematical descriptions They make good models for architectural and man made items They do not necessarily make good natural or organic items Blobby objects have no easy mathematical formula
Copyright © Curt Hill CSG Constructive (or Computed) Solid Geometry Build a complex solid with set operations on simple solids The primitives are rectanguloids, spheres, cones, cylinders and the like Union, intersection, set difference are the operations
Copyright © Curt Hill Deformation Take a solid and warp it in some way Deform a sphere into a head Use gravity field to smooth an object Morphing is a form of deformation
Copyright © Curt Hill Morphing Transform one image or solid into another Linear interpolation from control points in source image to that in the destination image Pretty common operation any more Hardware operation on SGI computers Quake also does a morph automatically in the animation
Copyright © Curt Hill QuadTrees and OctTrees QuadTree is for 2D and OctTree for 3D QuadTree described –Divide surface into 4 quarters –Evaluate each quarter –If the quarter is empty then it is done –Otherwise re-divide into four more –Quit when the image is acceptable OctTree divide 3 space into 8 quadrants –Basis of Binary Space Partitioning tree (BSP) –It is better if any planar surfaces are parallel to the divisions
Copyright © Curt Hill Fractals An algorithm not a formula Describes a process where an image or figure may be computed Usually recursive or highly iterative Two fractal web sites that demonstrate: – –sprott.physics.wisc.edu/fractals/sprott.physics.wisc.edu/fractals/
Copyright © Curt Hill Process to Make a Mountain Start with a tetrahedron/pyramid Take each of the showing faces and subdivide into several triangles –Randomly choose a midpoint that subdivides Randomly push or pull the middle vertex off the plane Recursively do this until satisfactory appearance
Copyright © Curt Hill What we learn from Legos Large irregular solids may be constructed from Lego-like pieces All we have to do is make sure that the basic block is small compared to the large object –Works for CSG and meshes of polygons or any other construction of large from small We can then smooth the object with curves
Copyright © Curt Hill Smoothing Numerous spline function for making series of points into a curve Cubic spline –Take first four points –Find the cubic that goes through these –Move up one point and find the next cubic –Each cubic only does one segment Bezier curves are easier to compute but also create a curve
Copyright © Curt Hill Rendering The process of displaying in a 3D perspective Must do ray tracing Shading Complicated (that is slowed) by: –Large numbers of objects or polygons –Layers that modify the light path –Reflections
Copyright © Curt Hill VTK Visualization Toolkit is an open source package for visualization – It comes with a ton of C functions for building and manipulating 3D objects It also comes with a 3D renderer Do a demo