By John Vince

A entire evaluate of the geometry linked to special effects that offers every thing a reader must comprehend the topic.

Includes a precis hundreds and hundreds of formulae used to unravel second and 3D geometric difficulties; labored examples; proofs; mathematical techniques for fixing geometric difficulties; a thesaurus of phrases utilized in geometry.

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Extra resources for Geometry for Computer Graphics: Formulae, Examples and Proofs

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1. 2. 1 houses of circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 2. 2 Ellipses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nine nine 10 1. three Triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. three. 1 kinds of triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. three. 2 comparable triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. three. three Congruent triangles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. three. four Theorem of Pythagoras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. three. five inner and exterior angles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. three. 6 Sine, cosine and tangent principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. three. 7 zone of a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. three. eight Inscribed and circumscribed circles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. three. nine Centroid of a triangle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. three. 10 round trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . eleven eleven eleven 12 12 thirteen thirteen thirteen 14 15 15 1. four Quadrilaterals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sixteen 1. five Polygons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. five. 1 inner and exterior angles of a polygon . . . . . . . . . . . . . . . . . . . . . . . . . 1. five. 2 trade inner angles of a cyclic polygon . . . . . . . . . . . . . . . . . . . . . . 1. five. three zone of a standard polygon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 19 19 19 1. 6 third-dimensional gadgets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 6. 1 Prisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 6. 2 Pyramids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 6. three Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 6. four Cones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 6. five Spheres . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 21 21 22 22 22 xi xii Contents 1. 6. 6 1. 6. 7 Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Platonic solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 23 1. 7 Coordinate platforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 7. 1 Cartesian coordinates in ‫ޒ‬2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 7. 2 Cartesian coordinates in ‫ޒ‬3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 7. three Polar coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 7. four Cylindrical coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1. 7. five round coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 26 26 27 27 28 1. eight Vectors 1. eight. 1 1. eight. 2 1. eight. three 1. eight. four 1. eight. five 1. eight. 6 1. eight. 7 1. eight. eight 1. eight. nine 1. eight. 10 1. eight. eleven 1. eight. 12 1. eight. thirteen 1. eight. 14 1. eight. 15 1. eight. sixteen 1. eight. 17 1. eight. 18 ................................................................. Vector among issues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scaling a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reversing a vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unit Cartesian vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic notation for a vector . . . . . . .

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