(****************************************************************************** * Ctm3d * ******************************************************************************) Unit Ctm3d; (******************************************************************************* * Homogeneous Coordinates * * ----------------------- * * * * Homogeneous coordinates allow transformations to be represented by * * matrices. A 3x3 matrix is used for 2D transformations, and a 4x4 matrix* * for 3D transformations. * * * * THIS MODULE IMPLEMENTS ONLY 3D TRANSFORMATIONS. * * * * in homogeneous coordination the point P(x,y,z) is represented as * * P(w*x, w*y, w*z, w) for any scale factor w!=0. * * in this module w == 1. * * * * Transformations: * * 1. translation * * [x, y, z] --> [x + Dx, y + Dy, z + Dz] * * * * Ú ¿ * * ³1 0 0 0³ * * T(Dx, Dy, Dz) = ³0 1 0 0³ * * ³0 0 1 0³ * * ³Dx Dy Dz 1³ * * À Ù * * 2. scaling * * [x, y, z] --> [Sx * x, Sy * y, Sz * z] * * * * Ú ¿ * * ³Sx 0 0 0³ * * S(Sx, Sy) = ³0 Sy 0 0³ * * ³0 0 Sz 0³ * * ³0 0 0 1³ * * À Ù * * * * 3. rotation * * * * a) Around the Z axis: * * * * [x, y, z] --> [x*cost - t*sint, x*sint + y*cost, z] * * Ú ¿ * * ³cost sint 0 0³ * * Rz(t) = ³-sint cost 0 0³ * * ³0 0 1 0³ * * ³0 0 0 1³ * * À Ù * * * * b) Around the X axis: * * * * [x, y, z] --> [x, y*cost - z*sint, y*sint + z*cost] * * Ú ¿ * * ³1 0 0 0³ * * Rx(t) = ³0 cost sint 0³ * * ³0 -sint cost 0³ * * ³0 0 0 1³ * * À Ù * * * * c) Around the Y axis: * * * * [x, y, z] --> [xcost + z*sint, y, z*cost - x*sint] * * Ú ¿ * * ³cost 0 -sint 0³ * * Ry(t) = ³0 1 0 0³ * * ³sint 0 cost 0³ * * ³0 0 0 1³ * * À Ù * * * * transformation of the vector [x,y,z,1] by transformation matrix T is given * * by the formula: * * Ú ¿ * * [x', y', z', 1] = [x,y,z,1]³ T ³ * * À Ù * * Optimizations: * * The most general composition of R, S and T operations will produce a matrix* * of the form: * * Ú ¿ * * ³r11 r12 r13 0³ * * ³r21 r22 r23 0³ * * ³r31 r32 r33 0³ * * ³tx ty tz 1³ * * À Ù * * The task of matrix multiplication can be simplified by * * x' = x*r11 + y*r21 + z*r31 + tx * * y' = x*r12 + y*r22 + z*r32 + ty * * z' = x*r13 + y*r23 + z*r33 + tz * * * * * * See also: * * "Fundamentals of Interactive Computer Graphics" J.D FOLEY A.VAN DAM * * Adison-Weslely ISBN 0-201-14468-9 pp 245-265 * *******************************************************************************) interface uses hdr3d; type ctmPtr = ^ctm; ctm = object r11, r12, r13 : real; r21, r22, r23 : real; r31, r32, r33 : real; tx, ty, tz : real; constructor SetUnit; constructor Copy(var src : ctm); procedure save(var dest : ctm); procedure translate(Dx, Dy, Dz : real); procedure translateX(dx : real); procedure translateY(dy : real); procedure translateZ(dz : real); {use these routines for single axis translations, they are faster!} procedure rotateX(t : real); procedure rotateY(t : real); procedure rotateZ(t : real); procedure scale(Sx, Sy, Sz : real); procedure scaleX(sx : real); procedure scaleY(sy : real); procedure scaleZ(sz : real); {use these routines for single axis scaling, they are faster!!!} procedure Left_translate(Dx, Dy, Dz : real); procedure Left_translateX(dx : real); procedure Left_translateY(dy : real); procedure Left_translateZ(dz : real); {use theseLeft_ routines for single axis translations, they are faster!} procedure Left_rotateX(t : real); procedure Left_rotateY(t : real); procedure Left_rotateZ(t : real); procedure Left_scale(Sx, Sy, Sz : real); procedure Left_scaleX(sx : real); procedure Left_scaleY(sy : real); procedure Left_scaleZ(sz : real); {use these routines for single axis scaling, they are faster!!!} procedure transform(var t: point3d; p : point3d); { Note that t (target) is var, but p is NOT. p cannot be a var parameter, because if the same point is transformed, data should be copied for correct results } procedure inv_transform(var p : point3d); { inv_transform changes the input point } procedure inverse; { M ^-1 } procedure multiply(var c : ctm); {multiply from right self * c} procedure Multiply_2(var a, b : ctm); end; implementation (*--------------------------------------------------------------------------- * ctm.SetUnit -- set the ctm to the unit matrix. *--------------------------------------------------------------------------- *) constructor ctm.SetUnit; begin r11 := 1; r12 := 0; r13 := 0; r21 := 0; r22 := 1; r23 := 0; r31 := 0; r32 := 0; r33 := 1; Tx := 0; Ty := 0; Tz := 0; end; (*--------------------------------------------------------------------------- * ctm.Copy -- Copy another ctm to self. *--------------------------------------------------------------------------- *) constructor ctm.copy; begin { This copy is done this way instead of Self := Src; becuase TP does not do that as expected. } r11 := Src.r11; r12 := Src.r12; r13 := Src.r13; r21 := Src.r21; r22 := Src.r22; r23 := Src.r23; r31 := Src.r31; r32 := Src.r32; r33 := Src.r33; tx := Src.tx; ty := Src.ty; tz := Src.tz; end; (* ctm.save save ctm in a specified variable ctm *) (******************************************************************************* * ctm.save * *******************************************************************************) procedure ctm.save; begin dest := self; end; (*--------------------------------------------------------------------------- * ctm.translate -- multyply ctm by T(dx,dy,dz). ctm = ctm * T(dx,dy,dz) * Ú ¿ Ú ¿ Ú ¿ * ³r11 r12 r13 0³ ³1 0 0 0³ ³r11 r12 r31 0³ * ³r21 r22 r23 0³ * ³0 1 0 0³ = ³r21 r22 r31 0³ * ³r31 r32 r33 0³ ³0 0 1 0³ ³r21 r22 r31 0³ * ³tx ty tz 1³ ³Dx Dy Dz 1³ ³tx' ty' tz' 1³ * À Ù À Ù À Ù * * tx' = tx + Dx * ty' = ty + Dy * tz' = tz + Dz *--------------------------------------------------------------------------- *) procedure ctm.translate; begin Tx := Tx + Dx; Ty := Ty + Dy; Tz := Tz + Dz; end; (*--------------------------------------------------------------------------- * ctm.Left_translate -- multyply ctm by T(dx,dy,dz) on the left. * ctm = T(dx,dy,dz) * ctm * Ú ¿ Ú ¿ Ú ¿ * ³1 0 0 0³ ³r11 r12 r13 0³ ³r11 r12 r31 0³ * ³0 1 0 0³ ³r21 r22 r23 0³ = ³r21 r22 r31 0³ * ³0 0 1 0³ ³r31 r32 r33 0³ ³r21 r22 r31 0³ * ³Dx Dy Dz 1³ ³tx ty tz 1³ ³tx' ty' tz' 1³ * À Ù À Ù À Ù * * tx' = r11 * Dx + r21 * Dy + r31 * Dz + tx * ty' = r12 * Dx + r22 * Dy + r32 * Dz + ty * tz' = r13 * Dx + r23 * Dy + r33 * Dz + tz *--------------------------------------------------------------------------- *) procedure ctm.Left_translate; begin Tx := Tx + r11 * Dx + r21 * Dy + r31 * Dz; Ty := Ty + r12 * Dx + r22 * Dy + r32 * Dz; Tz := Tz + r13 * Dx + r23 * Dy + r33 * Dz; end; (******************************************************************************* * ctm.translateX * *******************************************************************************) procedure ctm.translateX; begin tx := tx+dx; end; (******************************************************************************* * ctm.Left_translateX * *******************************************************************************) procedure ctm.Left_translateX; begin tx := tx + dx * r11; ty := ty + dx * r12; tz := tz + dx * r13; end; (******************************************************************************* * ctm.translateY * *******************************************************************************) procedure ctm.translateY; begin ty := ty+dy; end; (******************************************************************************* * ctm.Left_translateY * *******************************************************************************) procedure ctm.Left_translateY; begin tx := tx + dy * r21; ty := ty + dy * r22; tz := tz + dy * r23; end; (******************************************************************************* * ctm.translateZ * *******************************************************************************) procedure ctm.translateZ; begin tz := tz+dz; end; (******************************************************************************* * ctm.Left_translateZ * *******************************************************************************) procedure ctm.Left_translateZ; begin tx := tx + dz * r31; ty := ty + dz * r32; tz := tz + dz * r33; end; {the above 6 procedures are used for single axis translation - # of FP operations is 1/3, # of arguments passed is 1/3, try using these procedures when possiable to improve animation} (*--------------------------------------------------------------------------- * ctm.scale -- multyply ctm by S(sx,sy,sz). ctm = ctm * S(sx,sy,sz) * Ú ¿ Ú ¿ Ú ¿ * ³r11 r12 r13 0³ ³Sx 0 0 0³ ³Sx*r11 Sy*r12 Sz*r13 0³ * ³r21 r22 r23 0³ * ³0 Sy 0 0³ = ³Sx*r21 Sy*r22 Sz*r13 0³ * ³r31 r32 r33 0³ ³0 0 Sz 0³ ³Sx*r31 Sy*r32 Sz*r33 0³ * ³tx ty tz 1³ ³0 0 0 1³ ³Sx*tx Sy*ty Sz*tz 1³ * À Ù À Ù À Ù *--------------------------------------------------------------------------- *) procedure ctm.scale; begin r11 := r11*Sx; r12 := r12*Sy; r13 := r13*Sz; r21 := r21*Sx; r22 := r22*Sy; r23 := r23*Sz; r31 := r31*Sx; r32 := r32*Sy; r33 := r33*Sz; tx := tx*Sx; ty := ty*Sy; tz := tz*Sz end; (*--------------------------------------------------------------------------- * ctm.Left_scale -- multyply ctm by S(sx,sy,sz) on the left. * ctm = S(sx,sy,sz) * ctm * Ú ¿ Ú ¿ Ú ¿ * ³Sx 0 0 0³ ³r11 r12 r13 0³ ³Sx*r11 Sy*r12 Sz*r13 0³ * ³0 Sy 0 0³ * ³r21 r22 r23 0³ = ³Sx*r21 Sy*r22 Sz*r13 0³ * ³0 0 Sz 0³ ³r31 r32 r33 0³ ³Sx*r31 Sy*r32 Sz*r33 0³ * ³0 0 0 1³ ³tx ty tz 1³ ³ tx ty tz 1³ * À Ù À Ù À Ù *--------------------------------------------------------------------------- *) procedure ctm.Left_scale; begin r11 := r11*Sx; r12 := r12*Sy; r13 := r13*Sz; r21 := r21*Sx; r22 := r22*Sy; r23 := r23*Sz; r31 := r31*Sx; r32 := r32*Sy; r33 := r33*Sz; end; (******************************************************************************* * ctm.scaleZ * *******************************************************************************) procedure ctm.scaleZ; begin r13 := r13*Sz; r23 := r23*Sz; r33 := r33*Sz; tz := tz*Sz; end; (******************************************************************************* * ctm.Left_scaleZ * *******************************************************************************) procedure ctm.Left_scaleZ; begin r31 := r31*Sz; r32 := r32*Sz; r33 := r33*Sz; end; (******************************************************************************* * ctm.scaleY * *******************************************************************************) procedure ctm.scaleY; begin r12 := r12*Sy; r22 := r22*Sy; r32 := r32*Sy; ty := ty*Sy; end; (******************************************************************************* * ctm.Left_scaleY * *******************************************************************************) procedure ctm.Left_scaleY; begin r21 := r21*Sy; r22 := r22*Sy; r23 := r23*Sy; end; (******************************************************************************* * ctm.scaleX * *******************************************************************************) procedure ctm.scaleX; begin r11 := r11*Sx; r21 := r21*Sx; r31 := r31*Sx; tx := tx*Sx; end; (******************************************************************************* * ctm.Left_scaleX * *******************************************************************************) procedure ctm.Left_scaleX; begin r11 := r11*Sx; r12 := r12*Sx; r13 := r13*Sx; end; {the above 6 routines should be used for single axis scale, they are much faster then calling the general scale routine (with a factor of better then 3), and require less arguments} (*--------------------------------------------------------------------------- * ctm.rotateZ -- multyply ctm by Rz(t). ctm = ctm * Rz(t) * Ú ¿ Ú ¿ Ú ¿ * ³r11 r12 r13 0³ ³cost sint 0 0³ ³r11' r12' r13 0³ * ³r21 r22 r23 0³ ³-sint cost 0 0³ ³r21' r22' r23 0³ * ³r31 r32 r33 0³ * ³0 0 1 0³ = ³r31' r32' r33 0³ * ³tx ty tz 1³ ³0 0 0 1³ ³tx' ty' tz 1³ * À Ù À Ù À Ù * * r11' = r11*cost - r12*sint * r21' = r21*cost - r22*sint * r31' = r31*cost - r32*sint * tx' = tx *cost - ty *sint * r12' = r11*sint + r12*cost * r22' = r21*sint + r22*cost * r32' = r31*sint + r32*cost * ty' = tx *sint + ty *cost * *--------------------------------------------------------------------------- *) procedure ctm.rotateZ; var cost, sint : real; tmp : real; begin cost := cos(t / 180.0 * 3.1415926535897932385); sint := sin(t / 180.0 * 3.1415926535897932385); tmp := r11*cost - r12*sint; r12 := r11*sint + r12*cost; r11 := tmp; tmp := r21*cost - r22*sint; r22 := r21*sint + r22*cost; r21 := tmp; tmp := r31*cost - r32*sint; r32 := r31*sint + r32*cost; r31 := tmp; tmp := tx *cost - ty *sint; ty := tx *sint + ty *cost; tx := tmp; end; (*--------------------------------------------------------------------------- * ctm.Left_rotateZ -- multyply ctm by Rz(t) on the left. ctm = Rz(t) * ctm * Ú ¿ Ú ¿ Ú ¿ * ³cost sint 0 0³ ³r11 r12 r13 0³ ³r11' r12' r13' 0³ * ³-sint cost 0 0³ ³r21 r22 r23 0³ ³r21' r22' r23' 0³ * ³0 0 1 0³ * ³r31 r32 r33 0³ = ³r31 r32 r33 0³ * ³0 0 0 1³ ³tx ty tz 1³ ³tx ty tz 1³ * À Ù À Ù À Ù * * r11' = r11*cost + r21*sint * r12' = r12*cost + r22*sint * r13' = r13*cost + r23*sint * r21' = r21*cost - r11*sint * r22' = r22*cost - r12*sint * r23' = r23*cost - r13*sint * *--------------------------------------------------------------------------- *) procedure ctm.Left_rotateZ; var cost, sint : real; tmp : real; begin cost := cos(t / 180.0 * 3.1415926535897932385); sint := sin(t / 180.0 * 3.1415926535897932385); tmp := r11 * cost + r21 * sint; r21 := r21 * cost - r11 * sint; r11 := tmp; tmp := r12 * cost + r22 * sint; r22 := r22 * cost - r12 * sint; r12 := tmp; tmp := r13 * cost + r23 * sint; r23 := r23 * cost - r13 * sint; r13 := tmp; end; (*--------------------------------------------------------------------------- * ctm.rotateX -- multyply ctm by Rx(t). ctm = ctm * Rx(t) * Ú ¿ Ú ¿ Ú ¿ * ³r11 r12 r13 0³ ³1 0 0 0³ ³r11 r12' r13' 0³ * ³r21 r22 r23 0³ ³0 cost sint 0³ ³r21 r22' r23' 0³ * ³r31 r32 r33 0³ * ³0 -sint cost 0³ = ³r31 r32' r33' 0³ * ³tx ty tz 1³ ³0 0 0 1³ ³tx ty' tz' 1³ * À Ù À Ù À Ù * * r12' = r12*cost - r13*sint * r13' = r12*sint + r13*cost * r22' = r22*cost - r23*sint * r23' = r22*sint + r23*cost * ty' = ty *cost - tz *sint * tz' = ty *sint + tz *cost *--------------------------------------------------------------------------- *) procedure ctm.rotateX; var cost, sint : real; tmp : real; begin cost := cos(t / 180.0 * 3.1415926535897932385); {constants are evaluated at compile time} sint := sin(t / 180.0 * 3.1415926535897932385); tmp := r12*cost - r13*sint; r13 := r12*sint + r13*cost; r12 := tmp; tmp := r22*cost - r23*sint; r23 := r22*sint + r23*cost; r22 := tmp; tmp := r32*cost - r33*sint; r33 := r32*sint + r33*cost; r32 := tmp; tmp := ty *cost - tz *sint; tz := ty *sint + tz *cost; ty := tmp; end; (*--------------------------------------------------------------------------- * ctm.Left_rotateX -- multyply ctm by Rx(t) on the left. ctm = Rx(t) * ctm * Ú ¿ Ú ¿ Ú ¿ * ³1 0 0 0³ ³r11 r12 r13 0³ ³r11 r12 r13 0³ * ³0 cost sint 0³ ³r21 r22 r23 0³ ³r21' r22' r23' 0³ * ³0 -sint cost 0³ * ³r31 r32 r33 0³ = ³r31' r32' r33' 0³ * ³0 0 0 1³ ³tx ty tz 1³ ³tx ty tz 1³ * À Ù À Ù À Ù * * r21' = r21*cost + r31*sint * r22' = r22*cost + r32*sint * r23' = r23*cost + r33*sint * r31' = r31*cost - r21*sint * r32' = r32*cost - r22*sint * r33' = r33*cost - r23*sint * *--------------------------------------------------------------------------- *) procedure ctm.Left_rotateX; var cost, sint : real; tmp : real; begin cost := cos(t / 180.0 * 3.1415926535897932385); sint := sin(t / 180.0 * 3.1415926535897932385); tmp := r21 * cost + r31 * sint; r31 := r31 * cost - r21 * sint; r21 := tmp; tmp := r22 * cost + r32 * sint; r32 := r32 * cost - r22 * sint; r22 := tmp; tmp := r23 * cost + r33 * sint; r33 := r33 * cost - r23 * sint; r23 := tmp; end; (*--------------------------------------------------------------------------- * ctm.rotateY -- multyply ctm by Rx(t). ctm = ctm * Ry(t) * Ú ¿ Ú ¿ Ú ¿ * ³r11 r12 r13 0³ ³cost 0 -sint 0³ ³r11' r12 r13' 0³ * ³r21 r22 r23 0³ ³0 1 0 0³ ³r21' r22 r23' 0³ * ³r31 r32 r33 0³ * ³sint 0 cost 0³ = ³r31' r32 r33' 0³ * ³tx ty tz 1³ ³0 0 0 1³ ³tx ty' tz' 1³ * À Ù À Ù À Ù * * r11' = r11*cost + r13*sint * r13' = r13*cost - r11*sint * r21' = r21*cost + r23*sint * r23' = r23*cost - r21*sint * tx' = tx *cost + tz *sint * tz' = tz *cost - tx *sint *--------------------------------------------------------------------------- *) procedure ctm.rotateY; var cost, sint : real; tmp : real; begin cost := cos(t / 180.0 * 3.1415926535897932385); sint := sin(t / 180.0 * 3.1415926535897932385); tmp := r11*cost + r13*sint; r13 := r13*cost - r11*sint; r11 := tmp; tmp := r21*cost + r23*sint; r23 := r23*cost - r21*sint; r21 := tmp; tmp := r31*cost + r33*sint; r33 := r33*cost - r31*sint; r31 := tmp; tmp := tx *cost + tz *sint; tz := tz *cost - tx *sint; tx := tmp; end; (*--------------------------------------------------------------------------- * ctm.Left_rotateY -- multyply ctm by Ry(t) on the left. ctm = Ry(t) * ctm * Ú ¿ Ú ¿ Ú ¿ * ³cost 0 -sint 0³ ³r11 r12 r13 0³ ³r11' r12' r13' 0³ * ³0 1 0 0³ ³r21 r22 r23 0³ ³r21 r22 r23 0³ * ³sint 0 cost 0³ * ³r31 r32 r33 0³ = ³r31' r32' r33' 0³ * ³0 0 0 1³ ³tx ty tz 1³ ³tx ty tz 1³ * À Ù À Ù À Ù * * r11' = r11*cost - r31*sint * r12' = r11*cost - r32*sint * r13' = r11*cost - r33*sint * r31' = r31*cost + r11*sint * r32' = r32*cost + r12*sint * r33' = r33*cost + r13*sint * *--------------------------------------------------------------------------- *) procedure ctm.Left_rotateY; var cost, sint : real; tmp : real; begin cost := cos(t / 180.0 * 3.1415926535897932385); sint := sin(t / 180.0 * 3.1415926535897932385); tmp := r11 * cost - r31 * sint; r31 := r31 * cost + r11 * sint; r11 := tmp; tmp := r11 * cost - r32 * sint; r32 := r32 * cost + r12 * sint; r12 := tmp; tmp := r11 * cost - r33 * sint; r33 := r33 * cost + r13 * sint; r13 := tmp; end; (*--------------------------------------------------------------------------- * ctm.transform -- transform [x,y,z] by the CTM. * * See remarks at the top. *--------------------------------------------------------------------------- *) procedure ctm.transform; begin t.x := p.x*r11 + p.y*r21 + p.z*r31 + tx; t.y := p.x*r12 + p.y*r22 + p.z*r32 + ty; t.z := p.x*r13 + p.y*r23 + p.z*r33 + tz; end; (*--------------------------------------------------------------------------- * ctm.inv_transform -- transform [x',y',z'] by the INVERS * transfomation CTM^-1 * * The inverse transformation is calculated by solving the 3 equations * for x,y,z. * * x' = x*r11 + y*r21 + z*r31 + tx * y' = x*r12 + y*r22 + z*r32 + ty * z' = x*r13 + y*r23 + z*r33 + tz * *--------------------------------------------------------------------------- *) procedure ctm.inv_transform; var d : real; { Delta } d_x : real; { Delta X [X = Delta X / Delta] } d_y : real; { Delta Y [Y = Delta Y / Delta] } d_z : real; { Delta Z [Z = Delta Z / Delta] } begin d := r11*(r22*r33 - r32*r23) -r21*(r12*r33 - r13*r32) +r31*(r12*r23 - r22*r13); d_x:= (p.x - tx)*(r22*r33 - r32*r23) -r21*((p.y - ty)*r33 - (p.z - tz)*r32) +r31*((p.y - ty)*r23 - r22*(p.z - tz)); d_y:= r11*((p.y - ty)*r33 - r32*(p.z - tz)) -(p.x - tx)*(r12*r33 - r13*r32) +r31*(r12*(p.z - tz) - (p.y - ty)*r13); d_z:= r11*(r22*(p.z - tz) - (p.y - ty)*r23) -r21*(r12*(p.z - tz) - r13*(p.y - ty)) +(p.x - tx)*(r12*r23 - r22*r13); p.x := round(d_x / d); p.y := round(d_y / d); p.z := round(d_z / d); end; (******************************************************************************* * ctm.multiply * * here we multiply our ctm with another from the right : self * c -> self * *******************************************************************************) procedure ctm.multiply; var t : ctm; begin t.r11 := r11*c.r11+r12*c.r21+r13*c.r31; t.r21 := r21*c.r11+r22*c.r21+r23*c.r31; t.r31 := r31*c.r11+r32*c.r21+r33*c.r31; t.tx := tx *c.r11+ty *c.r21+tz *c.r31+c.tx; t.r12 := r11*c.r12+r12*c.r22+r13*c.r32; t.r22 := r21*c.r12+r22*c.r22+r23*c.r32; t.r32 := r31*c.r12+r32*c.r22+r33*c.r32; t.ty := tx *c.r12+ty *c.r22+tz *c.r32+c.ty; t.r13 := r11*c.r13+r12*c.r23+r13*c.r33; t.r23 := r21*c.r13+r22*c.r23+r23*c.r33; t.r33 := r31*c.r13+r32*c.r23+r33*c.r33; t.tz := tx *c.r13+ty *c.r23+tz *c.r33+c.tz; copy(t); end; {ctm.multiply} (******************************************************************************* * ctm.multiply_2 * * here we multiply our ctm with another from the right : self * c -> self * *******************************************************************************) procedure ctm.multiply_2; begin r11 := a.r11*b.r11+a.r12*b.r21+a.r13*b.r31; r21 := a.r21*b.r11+a.r22*b.r21+a.r23*b.r31; r31 := a.r31*b.r11+a.r32*b.r21+a.r33*b.r31; tx := a.tx *b.r11+a.ty *b.r21+a.tz *b.r31+b.tx; r12 := a.r11*b.r12+a.r12*b.r22+a.r13*b.r32; r22 := a.r21*b.r12+a.r22*b.r22+a.r23*b.r32; r32 := a.r31*b.r12+a.r32*b.r22+a.r33*b.r32; ty := a.tx *b.r12+a.ty *b.r22+a.tz *b.r32+b.ty; r13 := a.r11*b.r13+a.r12*b.r23+a.r13*b.r33; r23 := a.r21*b.r13+a.r22*b.r23+a.r23*b.r33; r33 := a.r31*b.r13+a.r32*b.r23+a.r33*b.r33; tz := a.tx *b.r13+a.ty *b.r23+a.tz *b.r33+b.tz; end; {ctm.multiply_2} (******************************************************************************* * ctm.inverse * *******************************************************************************) procedure ctm.inverse; begin runerror; end; end.