Reading. Hierarchical Modeling. Symbols and instances. Required: Angel, sections , 9.8. Optional:

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1 Reading Required: Angel, sections , 9.8 Optional: Hierarchical Modeling OpenGL rogramming Guide, the Red Book, chapter 3 cse hierarchical 1 cse hierarchical 2 Symbols and instances Most graphics AIs support a few geometric primitives: spheres, cubes, cylinders these procedures define points for you, but they're still just points These symbols are instanced using an instance transformation. the points are originally defined in a local coordinate system (eg, unit cube) TRS RS S translated rotated scaled points to draw Q: What is the matrix for the instance transformation above? cse hierarchical 3 cse hierarchical 4

2 Connecting primitives 3D Example: A robot Consider this robot with 3 degrees of freedom: Base rotates about its vertical axis by θ Lower rotates in its xy-plane by φ Upper rotates in its xy-plane by ψ Lower Upper h 1 h 2 h 3 Base Q: What matrix do we use to transform the base? Q: What matrix for the upper? Q: What matrix for the lower? cse hierarchical 5 cse hierarchical 6 Robot implementation The robot can be displayed by keeping a global matrix and computing it at each step: Matrix M_model; main() robot_(); robot_() M_model = R_y(theta); base(); M_model = R_y(theta)*T(0,h1,0)*R_z(phi); lower_(); M_model = R_y(theta)*T(0,h1,0)*R_z(phi) *T(0,h2,0)*R_z(psi); upper_(); Do the matrix computations seem a tad wasteful? cse hierarchical 7 cse hierarchical 8

3 Robot implementation, better Instead of recalculating the global matrix each time, we can just update it in place by concatenating matrices on the right: Matrix M_model; main() M_model = Identity(); robot_(); robot_() M_model *= R_y(theta); base(); M_model *= T(0,h1,0)*R_z(phi); lower_(); M_model *= T(0,h2,0)*R_z(psi); upper_(); cse hierarchical 9 Robot implementation, OpenGL OpenGL maintains a global state matrix called the model-view matrix, which is updated by concatenating matrices on the right. main() glmatrixmode( GL_MODELVIEW ); glloadidentity(); robot_(); robot_() glrotatef( theta, 0.0, 1.0, 0.0 ); base(); gltranslatef( 0.0, h1, 0.0 ); glrotatef( phi, 0.0, 0.0, 1.0 ); lower_(); gltranslatef( 0.0, h2, 0.0 ); glrotatef( psi, 0.0, 0.0, 1.0 ); upper_(); cse hierarchical 10 ObjectAxes.cpp Hierarchical modeling Hierarchical models can be composed of instances using trees or DAGs: edges contain geometric transformations nodes contain geometry (and possibly drawing attributes) How might we draw the tree for the robot? cse hierarchical 11 cse hierarchical 12

4 A complex example: human figure Human figure implementation, OpenGL head left upper left lower right upper right lower torso left upper left lower right upper right lower Q: What s the most sensible way to traverse this tree? figure() torso(); glushmatrix(); gltranslate(... ); glrotate(... ); head(); glopmatrix(); glushmatrix(); gltranslate(... ); glrotate(... ); left_upper_(); glushmatrix(); gltranslate(... ); glrotate(... ); left_lower_(); glopmatrix(); glopmatrix(); cse hierarchical 13 cse hierarchical 14 Order of transformations Global, fixed coordinate system Let s revisit the very first simple example in this lecture. To draw the transformed house, we would write OpenGL code like: glmatrixmode( GL_MODELVIEW ); glloadidentity(); gltranslatef(... ); glrotatef(... ); glscalef(... ); house(); One way to think of transformations is as movement of points in a global, fixed coordinate system Build the transformation matrix sequentially from left to right: T, then R, then S Then apply the transformation matrix to the object points: multiply all the points in by the composite matrix TRS this transformation takes the points from original to final positions TRS RS S Note that we are building the composite transformation matrix by starting from the left and postmultiplying each additional matrix translated rotated scaled points to draw cse hierarchical 15 cse hierarchical 16

5 Local, changing coordinate system Animation Another way to think of transformations is as affecting a local coordinate system that the primitive is eventually drawn in. local frame The above examples are called articulated models: rigid parts connected by joints They can be animated by specifying the joint angles (or other display parameters) as functions of time. translate rotate This is EXACTLY the same transformation as on the previous page, it's just how you look at it. scale Draw! cse hierarchical 17 cse hierarchical 18 Key-frame animation Scene graphs The most common method for character animation in production is key-frame animation. The idea of hierarchical modeling can be extended to an entire scene, encompassing: Each joint specified at various key frames (not necessarily the same as other joints) System does interpolation or in-betweening many different objects lights camera position This is called a scene tree or scene graph. Scene Camera Object1 Doing this well requires: Light1 Light2 Object2 Object3 A way of smoothly interpolating key frames: splines A good interactive system A lot of skill on the part of the animator cse hierarchical 19 cse hierarchical 20

6 Summary Here s what you should take home from this lecture: All the boldfaced terms. How primitives can be instanced and composed to create hierarchical models using geometric transforms. How the notion of a model tree or DAG can be extended to entire scenes. How OpenGL transformations can be used in hierarchical modeling. How keyframe animation works. cse hierarchical 21