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Hypershot : Fun with Hyperbolic Geometry

Hypershot : Fun with Hyperbolic Geometry. Praneet Sahgal. Modeling Hyperbolic Geometry. Upper Half-plane Model (Poincaré half-plane model) Poincaré Disk Model Klein Model Hyperboloid Model (Minkowski Model). Image Source: Wikipedia. Upper Half Plane Model. Say we have a complex plane

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Hypershot : Fun with Hyperbolic Geometry

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  1. Hypershot: Fun with Hyperbolic Geometry Praneet Sahgal

  2. Modeling Hyperbolic Geometry • Upper Half-plane Model (Poincaré half-plane model) • Poincaré Disk Model • Klein Model • Hyperboloid Model (Minkowski Model) Image Source: Wikipedia

  3. Upper Half Plane Model • Say we have a complex plane • We define the positive portion of the complex axis as hyperbolic space • We can prove that there are infinitely many parallel lines between two points on the real axis Image Source: Hyperbolic Geometry by James W. Anderson

  4. Poincaré Disk Model • Instead of confining ourselves to the upper half plane, we use the entire unit disk on the complex plane • Lines are arcs on the disc orthogonal to the boundary of the disk • The parallel axiom also holds here Image Source: http://www.ms.uky.edu/~droyster/courses/spring08/math6118/Classnotes/Chapter09.pdf

  5. Klein Model • Similar to the Poincaré disk model, except chords are used instead of arcs • The parallel axiom holds here, there are multiple chords that do not intersect Image Source: http://www.geom.uiuc.edu/~crobles/hyperbolic/hypr/modl/kb/

  6. Hyperboloid Model • Takes hyperbolic lines on the Poincaré disk (or Klein model) and maps them to a hyperboloid • This is a stereographic projection (preserves angles) • Maps a 2 dimensional disk to 3 dimensional space (maps n space to n+1 space) • Generalizes to higher dimensions Image Source: Wikipedia

  7. Motion in Hyperbolic Space • Translation in x, y, and z directions is not the same! Here are the transformation matrices: • To show things in 3D Euclidean space, we need 4D Hyperbolic space x-direction y-direction z-direction

  8. The Project • Create a system for firing projectiles in hyperbolic space, like a first person shooter • Provide a sandbox for understanding paths in hyperbolic space

  9. Demonstration

  10. Notable behavior • Objects in the center take a long time to move; the space in the center is bigger (see right)

  11. Techincal challenges • Applying the transformations for hyperbolic translation • LOTS of matrix multiplication • Firing objects out of the wand • Rotational transformation of a vector • Distributing among the Cube’s walls • Requires Syzygy vector (the data structure) • Hyperbolic viewing frustum

  12. Adding to the project • Multiple weapons (firing patterns that would show different behavior) • Collisions with stationary objects • Path tracing • Making sure wall distribution works… • 3D models for gun and target (?)

  13. References • http://mathworld.wolfram.com/EuclidsPostulates.html • Hyperbolic Geometry by James W. Anderson • http://mathworld.wolfram.com/EuclidsPostulates.html • http://www.math.ecnu.edu.cn/~lfzhou/others/cannon.pdf • http://www.geom.uiuc.edu/~crobles/hyperbolic/hypr/modl/kb/

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