Difference between revisions of "ApplyMatrix Command"

Latest revision as of 14:52, 19 June 2023

ApplyMatrix( <Matrix>, <Object> ): Transforms the object O so that point P of O is mapped to:

point M*P, if P is a 2D point and M is a 2 x 2 matrix

Example: Let M={{cos(π/2),-sin(π/2)}, {sin(π/2), cos(π/2)}} be the transformation matrix and u = (2,1) a given vector (object). ApplyMatrix(M,u) yields the vector u'=(-1,2), i.e. the result of a mathematically positive rotation by 90° of vector u.

point project(M*(x(P), y(P), 1)), if P is a 2D point and M a 3 x 3 matrix: project is a projection, mapping point (x, y, z) to (x/z, y/z).

Example: Let M={{1,1,0},{0,1,1},{1,0,1}} be a matrix and u=(2,1) a given vector. ApplyMatrix(M,u) yields vector u'=(1,0.67). In effect \begin{pmatrix}1&1&0\\ 0&1&1\\1&0&1 \end{pmatrix} \begin{pmatrix}2\\ 1\\1 \end{pmatrix} = \begin{pmatrix}3\\ 2\\3 \end{pmatrix}, and (3/3 = 1, 2/3 ≈ 0.67) (rounding to 2 decimal places)

point M*P, if P is a 3D point and M a 3 x 3 matrix
point N*P, if P is a 3D point and M a 2 x 2 matrix: the matrix N is the completion or order 3 of M: given M = \begin{pmatrix}a&b\\ c&d \end{pmatrix} then N = \begin{pmatrix}a&b&0\\ c&d&0\\0&0&1 \end{pmatrix}

Note: This command also works with quadrics and images: it will return "undefined" when the determinant of the transformation matrix is 0.

@@ Line 1: / Line 1: @@
-<noinclude>{{Manual Page|version=4.2}}</noinclude>
+<noinclude>{{Manual Page|version=5.0}}</noinclude>{{command|vector-matrix}}
-{{command|vector-matrix}}
+; ApplyMatrix( <[[Matrices|Matrix]]>, <[[Geometric Objects|Object]]> ): Transforms the object ''O'' so that point ''P'' of ''O'' is mapped to:
-; ApplyMatrix[ <[[Matrices|Matrix]]>, <[[Geometric Objects|Geometric Object]]>]: Transforms the object ''O'' so that point ''P'' of ''O'' is mapped to
+* point ''M*P'', if ''P'' is a ''2D'' point and ''M'' is a 2 x 2 matrix
-* point ''M*P'' (with matrix ''M'') in case M is a 2x2 matrix or
+:{{example|1= Let <code>M={{cos(π/2),-sin(π/2)}, {sin(π/2), cos(π/2)}}</code> be the transformation matrix and <code>u = (2,1)</code> a given vector (object). <code>ApplyMatrix(M,u)</code> yields the vector ''u'=(-1,2)'', i.e. the result of a mathematically positive rotation by 90° of vector ''u''.}}
-* point ''project(M*(x(P), y(P), 1))'' where ''project'' is a projection mapping point ''(x,y,z)'' to ''(x/z, y/z)'' in case of 3x3 matrix.
+* point ''project(M*(x(P), y(P), 1))'', if ''P'' is a ''2D'' point and ''M'' a 3 x 3 matrix: ''project'' is a projection, mapping point ''(x, y, z)'' to ''(x/z, y/z)''.
+:{{example|1=Let <code>M={{1,1,0},{0,1,1},{1,0,1}}</code> be a matrix and <code>u=(2,1)</code> a given vector. <code>ApplyMatrix(M,u)</code> yields vector ''u'=(1,0.67)''. In effect <math>\begin{pmatrix}1&1&0\\ 0&1&1\\1&0&1 \end{pmatrix}</math> <math>\begin{pmatrix}2\\ 1\\1 \end{pmatrix}</math> =  <math>\begin{pmatrix}3\\ 2\\3 \end{pmatrix}</math>, and (3/3 = 1, 2/3 ≈ 0.67) (rounding to 2 decimal places)}}
+* point ''M*P'', if ''P'' is a ''3D'' point and ''M'' a 3 x 3 matrix
-; ApplyMatrix[ <Matrix>, <[[Images|Image]]>]: Applies the same transformation as above to an image.
+* point ''N*P'', if  ''P'' is a ''3D'' point and ''M'' a 2 x 2 matrix: the matrix ''N'' is the ''completion or order 3'' of ''M'': given ''M'' = <math>\begin{pmatrix}a&b\\ c&d \end{pmatrix}</math> then ''N'' = <math>\begin{pmatrix}a&b&0\\ c&d&0\\0&0&1 \end{pmatrix}</math>
+:{{note| 1=This command also works with quadrics and [[Images|images]]: it will return "undefined" when the determinant of the transformation matrix is 0.}}

Difference between revisions of "ApplyMatrix Command"

Latest revision as of 14:52, 19 June 2023

ApplyMatrix

Command Categories (All commands)