VECTOR ANALYSIS FOR COMPUTER GRAPHICS
CONTENTS SCALARS AND VECTORS INTRODUCTION REPRESENTING VECTOR QUANTITIES NON-COLLINEAR VECTORS VECTOR REPRESENTATION CARTESIAN COORDINATES VECTOR NOTATION LENGTH OF A VECTOR VECTOR ALGEBRA UNIT VECTORS RECTANGULAR UNIT VECTORS POSITION VECTORS PROBLEM 1 PROBLEM 2 PROBLEM 3 VECTOR PRODUCTS SCALAR PRODUCT VECTOR PRODUCT SURFACE NORMALS THE ALGEBRA OF VECTOR PRODUCTS 2 10 TRIPLE PRODUCTS 2 10 1 SCALAR TRIPLE PRODUCT 2 10 2 THE VECTOR TRIPLE PRODUCT 2 11 PERPENDICULAR VECTORS 2 12 INTERPOLATING VECTORS 2 12 1 LINEAR INTERPOLATION 2 12 2 SPHERICAL INTERPOLATION 2 13 DIRECTION COSINES 2 14 CHANGE OF AXIAL SYSTEM 2 15 SUMMARY STRAIGHT LINES THE PARAMETRIC FORM OF THE LINE EQUATION THE CARTESIAN FORM OF THE LINE EQUATION THE GENERAL FORM OF THE LINE EQUATION 2D SPACE PARTITIONING A LINE PERPENDICULAR TO A VECTOR THE POSITION AND DISTANCE OF A POINT ON A LINE PERPENDICULAR TO THE ORIGIN THE POSITION AND DISTANCE OF THE NEAREST POINT ON A LINE TO A POINT THE POSITION OF A POINT REFLECTED IN A LINE 3 10 A LINE PERPENDICULAR TO A LINE THROUGH A POINT 3 10 1 THE PARAMETRIC FORM OF THE LINE EQUATION 3 11 A LINE EQUIDISTANT FROM TWO POINTS 3 12 THE EQUATION OF A LINE SEGMENT 3 13 THE INTERSECTION OF TWO STRAIGHT LINES 3 14 THE POINT OF INTERSECTION OF TWO 2D LINE SEGMENTS THE PLANE THE CARTESIAN FORM OF THE PLANE EQUATION THE PARAMETRIC FORM OF THE PLANE EQUATION A PLANE EQUATION FROM THREE POINTS A PLANE PERPENDICULAR TO A LINE AND PASSING THROUGH A POINT A PLANE THROUGH TWO POINTS AND PARALLEL TO A LINE 3D SPACE PARTITIONING THE ANGLE BETWEEN TWO PLANES THE ANGLE BETWEEN A LINE AND A PLANE 4 10 THE POSITION AND DISTANCE OF THE NEAREST POINT ON A PLANE TO A POINT 4 11 THE REFLECTION OF A POINT IN A PLANE 4 12 A PLANE BETWEEN TWO POINTS REFLECTIONS A LINE REFLECTING OFF A LINE A LINE REFLECTING OFF A PLANE INTERSECTIONS TWO INTERSECTING LINES IN R2 A LINE INTERSECTING A CIRCLE IN R2 A LINE INTERSECTING AN ELLIPSE IN R2 THE SHORTEST DISTANCE BETWEEN TWO SKEW LINES IN R3 TWO INTERSECTING LINES IN R3 A LINE INTERSECTING A PLANE A LINE INTERSECTING A SPHERE A LINE INTERSECTING AN ELLIPSOID 6 10 A LINE INTERSECTING A CYLINDER 6 11 A LINE INTERSECTING A CONE 6 12 A LINE INTERSECTING A TRIANGLE 6 13 A POINT INSIDE A TRIANGLE 6 14 A SPHERE INTERSECTING A PLANE 6 15 A SPHERE TOUCHING A TRIANGLE 6 16 TWO INTERSECTING PLANES ROTATING VECTORS ROTATING A VECTOR ABOUT AN ARBITRARY AXIS COMPLEX NUMBERS COMPLEX NUMBER OPERATIONS MAGNITUDE OF A COMPLEX NUMBER THE COMPLEX CONJUGATE GRAPHICAL INTERPRETATION OF COMPLEX NUMBERS I AS A ROTATOR UNIFYING E, I, SIN, AND COS COMPLEX NUMBERS AS ROTATORS QUATERNIONS QUATERNIONS AS ROTATORS THE COMPLEX CONJUGATE OF A QUATERNION THE NORM OF A QUATERNION INVERSE OF A QUATERNION ROTATING VECTORS USING QUATERNIONS 7 10 REPRESENTING A QUATERNION AS A MATRIX VECTOR DIFFERENTIATION THE DERIVATIVE OF A VECTOR THE NORMAL VECTOR TO A PLANAR CURVE THE NORMAL VECTOR TO A SURFACE PROJECTIONS PERSPECTIVE TRANSFORM HORIZONTALLY OBLIQUE PROJECTION PLANE VERTICALLY OBLIQUE PROJECTION PLANE ARBITRARY ORIENTATION OF THE PROJECTION PLANE PSEUDO FISH-EYE PROJECTION 10 RENDERING 10 1 INTRODUCTION 10 2 LIGHT SOURCES 10 3 LOCAL REFLECTION MODELS 10 4 SHADING 10 5 BUMP MAPPING 11 MOTION 11 1 INTRODUCTION 11 2 CLOSE ENCOUNTERS OF THE FIRST KIND 11 3 CLOSE ENCOUNTERS OF THE SECOND KIND REFERENCES