quaternion(9) [plan9 man page]
QUATERNION(9.2) QUATERNION(9.2) NAME
qtom, mtoq, qadd, qsub, qneg, qmul, qdiv, qunit, qinv, qlen, slerp, qmid, qsqrt - Quaternion arithmetic SYNOPSIS
#include <libg.h> #include <geometry.h> Quaternion qadd(Quaternion q, Quaternion r) Quaternion qsub(Quaternion q, Quaternion r) Quaternion qneg(Quaternion q) Quaternion qmul(Quaternion q, Quaternion r) Quaternion qdiv(Quaternion q, Quaternion r) Quaternion qinv(Quaternion q) double qlen(Quaternion p) Quaternion qunit(Quaternion q) void qtom(Matrix m, Quaternion q) Quaternion mtoq(Matrix mat) Quaternion slerp(Quaternion q, Quaternion r, double a) Quaternion qmid(Quaternion q, Quaternion r) Quaternion qsqrt(Quaternion q) DESCRIPTION
The Quaternions are a non-commutative extension field of the Real numbers, designed to do for rotations in 3-space what the complex numbers do for rotations in 2-space. Quaternions have a real component r and an imaginary vector component v=(i,j,k). Quaternions add component- wise and multiply according to the rule (r,v)(s,w)=(rs-v.w, rw+vs+vxw), where . and x are the ordinary vector dot and cross products. The multiplicative inverse of a non-zero quaternion (r,v) is (r,-v)/(r2-v.v). The following routines do arithmetic on quaternions, represented as typedef struct Quaternion Quaternion; struct Quaternion{ double r, i, j, k; }; Name Description qadd Add two quaternions. qsub Subtract two quaternions. qneg Negate a quaternion. qmul Multiply two quaternions. qdiv Divide two quaternions. qinv Return the multiplicative inverse of a quaternion. qlen Return sqrt(q.r*q.r+q.i*q.i+q.j*q.j+q.k*q.k), the length of a quaternion. qunit Return a unit quaternion (length=1) with components proportional to q's. A rotation by angle 0 about axis A (where A is a unit vector) can be represented by the unit quaternion q=(cos 0/2, Asin 0/2). The same rotation is represented by -q; a rotation by -0 about -A is the same as a rotation by 0 about A. The quaternion q transforms points by (0,x',y',z') = q-1(0,x,y,z)q. Quaternion multiplication composes rotations. The orientation of an object in 3-space can be represented by a quaternion giving its rotation relative to some `standard' orientation. The following routines operate on rotations or orientations represented as unit quaternions: mtoq Convert a rotation matrix (see tstack(9.2)) to a unit quaternion. qtom Convert a unit quaternion to a rotation matrix. slerp Spherical lerp. Interpolate between two orientations. The rotation that carries q to r is q-1r, so slerp(q, r, t) is q(q-1r)t. qmid slerp(q, r, .5) qsqrt The square root of q. This is just a rotation about the same axis by half the angle. SOURCE
/sys/src/libgeometry/quaternion.c SEE ALSO
tstack(9.2), qball(9.2) QUATERNION(9.2)
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MATRIX(9.2) MATRIX(9.2) NAME
ident, matmul, matmulr, determinant, adjoint, invertmat, xformpoint, xformpointd, xformplane, pushmat, popmat, rot, qrot, scale, move, xform, ixform, persp, look, viewport - Geometric transformations SYNOPSIS
#include <libg.h> #include <geometry.h> void ident(Matrix m) void matmul(Matrix a, Matrix b) void matmulr(Matrix a, Matrix b) double determinant(Matrix m) void adjoint(Matrix m, Matrix madj) double invertmat(Matrix m, Matrix inv) Point3 xformpoint(Point3 p, Space *to, Space *from) Point3 xformpointd(Point3 p, Space *to, Space *from) Point3 xformplane(Point3 p, Space *to, Space *from) Space *pushmat(Space *t) Space *popmat(Space *t) void rot(Space *t, double theta, int axis) void qrot(Space *t, Quaternion q) void scale(Space *t, double x, double y, double z) void move(Space *t, double x, double y, double z) void xform(Space *t, Matrix m) void ixform(Space *t, Matrix m, Matrix inv) int persp(Space *t, double fov, double n, double f) void look(Space *t, Point3 eye, Point3 look, Point3 up) void viewport(Space *t, Rectangle r, double aspect) DESCRIPTION
These routines manipulate 3-space affine and projective transformations, represented as 4x4 matrices, thus: typedef double Matrix[4][4]; Ident stores an identity matrix in its argument. Matmul stores axb in a. Matmulr stores bxa in b. Determinant returns the determinant of matrix m. Adjoint stores the adjoint (matrix of cofactors) of m in madj. Invertmat stores the inverse of matrix m in minv, returning m's determinant. Should m be singular (determinant zero), invertmat stores its adjoint in minv. The rest of the routines described here manipulate Spaces and transform Point3s. A Point3 is a point in three-space, represented by its homogeneous coordinates: typedef struct Point3 Point3; struct Point3{ double x, y, z, w; }; The homogeneous coordinates (x, y, z, w) represent the Euclidean point (x/w, y/w, z/w) if w!=0, and a ``point at infinity'' if w=0. A Space is just a data structure describing a coordinate system: typedef struct Space Space; struct Space{ Matrix t; Matrix tinv; Space *next; }; It contains a pair of transformation matrices and a pointer to the Space's parent. The matrices transform points to and from the ``root coordinate system,'' which is represented by a null Space pointer. Pushmat creates a new Space. Its argument is a pointer to the parent space. Its result is a newly allocated copy of the parent, but with its next pointer pointing at the parent. Popmat discards the Space that is its argument, returning a pointer to the stack. Nominally, these two functions define a stack of transformations, but pushmat can be called multiple times on the same Space multiple times, creating a transformation tree. Xformpoint and Xformpointd both transform points from the Space pointed to by from to the space pointed to by to. Either pointer may be null, indicating the root coordinate system. The difference between the two functions is that xformpointd divides x, y, z, and w by w, if w!=0, making (x, y, z) the Euclidean coordinates of the point. Xformplane transforms planes or normal vectors. A plane is specified by the coefficients (a, b, c, d) of its implicit equation ax+by+cz+d=0. Since this representation is dual to the homogeneous representation of points, libgeometry represents planes by Point3 structures, with (a, b, c, d) stored in (x, y, z, w). The remaining functions transform the coordinate system represented by a Space. Their Space * argument must be non-null -- you can't mod- ify the root Space. Rot rotates by angle theta (in radians) about the given axis, which must be one of XAXIS, YAXIS or ZAXIS. Qrot trans- forms by a rotation about an arbitrary axis, specified by Quaternion q. Scale scales the coordinate system by the given scale factors in the directions of the three axes. Move translates by the given displace- ment in the three axial directions. Xform transforms the coordinate system by the given Matrix. If the matrix's inverse is known a priori, calling ixform will save the work of recomputing it. Persp does a perspective transformation. The transformation maps the frustum with apex at the origin, central axis down the positive y axis, and apex angle fov and clipping planes y=n and y=f into the double-unit cube. The plane y=n maps to y'=-1, y=f maps to y'=1. Look does a view-pointing transformation. The eye point is moved to the origin. The line through the eye and look points is aligned with the y axis, and the plane containing the eye, look and up points is rotated into the x-y plane. Viewport maps the unit-cube window into the given screen viewport. The viewport rectangle r has r.min at the top left-hand corner, and r.max just outside the lower right-hand corner. Argument aspect is the aspect ratio (dx/dy) of the viewport's pixels (not of the whole viewport). The whole window is transformed to fit centered inside the viewport with equal slop on either top and bottom or left and right, depending on the viewport's aspect ratio. The window is viewed down the y axis, with x to the left and z up. The viewport has x increas- ing to the right and y increasing down. The window's y coordinates are mapped, unchanged, into the viewport's z coordinates. SOURCE
/sys/src/libgeometry/matrix.c MATRIX(9.2)