homology - Maple Help

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

 > $\mathrm{with}\left(\mathrm{geometry}\right):$
 > $\mathrm{point}\left(\mathrm{OO},0,0\right):$

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

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