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geometry

 StretchRotation
 find the stretch-rotation of a geometric object
 homology
 find the homology of a geometric object
 SpiralRotation
 find the spiral-rotation of a geometric object

 Calling Sequence StretchRotation(Q, P, O, theta, dir, k) homology(Q, P, O, theta, dir, k) SpiralRotation(Q, P, O, theta, dir, k)

Parameters

 Q - the name of the object to be created P - geometric object O - point which is the center of the homology theta - number which is the angle of the homology dir - name which is either clockwise or counterclockwise k - number which is the ratio of the homology

Description

 • Let O be a fixed point in the plane, k a given nonzero real number, theta and dir denote a given sensed angle. By the homology ( or stretch-rotation, or spiral-rotation) $H\left(\mathrm{O},k,\mathrm{\theta }\right)$ we mean the product $R\left(\mathrm{O},\mathrm{theta}\right)H\left(\mathrm{O},k\right)$ where $R\left(\mathrm{O},\mathrm{\theta },\mathrm{dir}\right)$ is the rotation with respect to O an angle theta in direction dir and $H\left(\mathrm{O},k\right)$ is the dilatation with respect to the center O and ratio k.
 • Point O is called the center of the homology, k the ratio of the homology, theta and dir the angle of the homology.
 • For a detailed description of Q (the object created), use the routine detail (i.e., detail(Q))
 • The command with(geometry,StretchRotation) allows the use of the abbreviated form of this command.

Examples

 > with(geometry):
 > point(OO,0,0):

define the parabola with vertex at (0,0) and focus at (0,1/2)

 > parabola(p1,['vertex'=point('ver',0,0),'focus'=point('fo',0,1/2)]):
 > Equation(p1,[x,y]);
 $\frac{{{x}}^{{2}}}{{4}}{-}\frac{{y}}{{2}}{=}{0}$ (1)
 > homology(p2,p1,OO,Pi/2,'counterclockwise',2):
 > Equation(p2);
 $\frac{{{y}}^{{2}}}{{16}}{+}\frac{{x}}{{4}}{=}{0}$ (2)
 > homology(p3,p1,OO,Pi,'counterclockwise',2):
 > Equation(p3);
 $\frac{{{x}}^{{2}}}{{16}}{+}\frac{{y}}{{4}}{=}{0}$ (3)
 > homology(p4,p1,OO,Pi/2,'clockwise',2):
 > Equation(p4);
 $\frac{{{y}}^{{2}}}{{16}}{-}\frac{{x}}{{4}}{=}{0}$ (4)
 > homology(p5,p1,OO,0,'clockwise',2):
 > Equation(p5);
 $\frac{{{x}}^{{2}}}{{16}}{-}\frac{{y}}{{4}}{=}{0}$ (5)
 > draw({p1(color=green,style=LINE,thickness=2,numpoints=50),p2,p3,p4,p5},    style=POINT,numpoints=200,color=brown,title=homology of a parabola);