Mark Nelson 3d projections Fall

The 3d pipeline (expansive view)  Tools stage  Asset conditioning stage  Application stage  | Geometry processing stage  | Rasterization stage

Tools stage  3d modeling  Export meshes (possibly w/ metadata)  Create textures

Asset conditioning stage  Platform- or engine-specific format conversations  Dependency resolution  ”Baked-in” effects  E.g., static lighting

Application stage  Run-time management in the engine  Prepare a scene  Combine e.g. Movable objects into one scene description  Omit anything that can’t possibly be visible  Set GPU rendering parameters

Basic GPU pipeline  Receive triangles  Triples of (x,y,z) vertices  Compute transformations  Rasterize  Turn into (x,y) screen pixels

World space  One 3d coordinate axis with all objects in a scene  Pre-culled by the engine to omit things that can’t possibly be visible  Constitutes the world geometry  E.g., can compute distances, collisions, etc.

Model space  We could have only world space  But, we often model objects externally (e.g. in 3dsmax)  Model space is the local coordinate space of one model, independent of a scene  Typically:  centered at (0,0,0)  aligned to an axis

Model to world space  To build a scene, all models have to be converted from local to world coordinates  Place in scene, then translate, rotate, and/or scale  Can be done ahead of time or on the GPU

Scene graph  Hierarchical data structure  Represents how to build a scene out of models  Root is world space  A transformation applies to anything below it in the tree  Can enable other optimizations

Scene graph

Camera  Engine and scene graph build up a scene description  In world space, from models in model space  We the viewer are somewhere in this world  At a coordinate (x,y,z)  Facing along a particular direction vector (x’,y’,z’)  What it looks like to us is view space

View space  In view space, we are:  at (0,0,0)  perpendicular to the (x,y) plane  facing along the z axis  Need to translate and rotate the world-space coordinates  3d version of rotating a map so up is where we’re facing

Projection  Project the (still 3d) view space onto our 2d screen  Orthographic projection  Just ignore z coordinate: (x,y,z)  (x,y) for all points  Perspective projection  Further away objects look smaller

Frustum

Perspective options  #1: First turn 3d view space into 3d perspective space  Make further away stuff smaller  Then later do an orthographic projection  Or, #2: Project directly  Impacts how things like frustum culling work

Simple perspective projection  If viewable depths are from z=1 to z=infinity:  x’ = x/z  y’ = y/z  2d screen centered at (0,0)

Wireframe projection  For each triangle  Project each vertex to 2d  Draw lines connecting them in 2d

Wireframe projection

Summary  Model space to world space  World space to view space  Projection  Missing: occlusion, lighting, shading

Transformation matrices  2d rotation  As matrix:

Transformation matrices  3d rotation is analogous  Can also do: scaling, shearing  However, translation can’t be directly done as a matrix  x’ = x + x_offset  y’ = y + y_offset  No matrix-multiply equivalent

Homogeneous coordinates  Extend 3d points and vectors to a 4d space  Stand-in dimension w=1  Now can define a translation transform as well  So all basic transforms can be chained  Get back to 3d by dividing x/y/z by w

Translation in matrix form

Affine transformations  Can represent all the relevant transformations with homogeneous coordinate 4x4 transform matrices  Translation, rotation, scaling, perspective transform  Common way of representing any transformation in APIs  Advanced alternative: quaternions

Project 2: a DIY renderer  Wireframe renderer  Due 22 October  Input: 3d coordinates, view position, view direction  Project to 2d coordinates, and draw (to screen or image)  Tuesday: more on perspective, and surfaces