PolarRectangles Package

PolarRectangle[{rmin,rmax},{thetamin,thetamax}] will produce a Graphics object consisting of the polar 'rectangle' with rmin < r < rmax and thetamin < theta < thetamax. Use the Show command to display the graphic. The option RectangleStyle may be set to any Graphics directive (such as Hue[1] or Thickness[.01]), or to any list of such directives. PolarRectangle will also accept any Graphics options.

PolarRectangle3D[{rmin,rmax},{thetamin,thetamax},h] will produce a Graphics3D object consisting of the polar 'rectangle' in the plane z=h with rmin < r < rmax and thetamin < theta < thetamax. Use the Show command to display the graphic. The option RectangleStyle may be set to any Graphics3D directive (such as Hue[1] or Thickness[.01]), or to any list of such directives. PolarRectangle3D will also accept any Graphics3D options.

PolarGrid[{rmin,rmax},{thetamin,thetamax}] will produce a Graphics object consisting of the 10 by 10 polar grid with rmin < r < rmax and thetamin < theta < thetamax. Use the Show command to display the graphic. PolarGrid[{rmin,rmax,dr},{thetamin,thetamax,dtheta}] will produce the grid with radial increments of dr and angular increments of d theta. The option GridStyle may be set to any Graphics directive (such as Hue[1] or Thickness[.01]), or to any list of such directives. PolarGrid will also accept any Graphics options.

PolarGrid3D[{rmin,rmax},{thetamin,thetamax}] will produce a Graphics3D object consisting of the 10 by 10 polar grid in the plane z=0 with rmin < r < rmax and thetamin < theta < thetamax. Use the Show command to display the graphic. PolarGrid3D[{rmin,rmax,dr},{thetamin,thetamax,d theta}] will produce the grid with radial increments of dr and angular incements of dtheta. The option GridStyle may be set to any Graphics3D directive (such as Hue[1] or Thickness[.01]), or to any list of such directives. PolarGrid3D will also accept any Graphics3D options.

PolarSolid[{rmin,rmax}, {thetamin,thetamax},{h1,h2}] will produce a Graphics3D object comprised of the polygons that make up the 'rectangular' solid from height z=h1 to height z=h2 over the polar 'rectangle' with rmin < r < rmax and thetamin < theta < thetamax. To view it, use the command Show[PolarSolid[...]]. The option ArcSegments may be set to any positive integer value greater than 1; it controls the number of line segments used in representing the curved portions of the top and bottom polar rectangles. PolarSolid will also accept any Graphics3D options.

PolarRiemannSum[f,{r,rmin,rmax},{theta, thetamin,thetamax}] produces a graphic showing side by side the polar rectangular solids in a Riemann sum, and a graph of the function f[r,theta] over the polar grid in the x y plane. PolarRiemannSum[f,{r,rmin,rmax,dr},{theta,thetamin, thetamax,dtheta}] will produce the Riemann sum with radial increments of dr and angular incements of dtheta. The option ArcSegments controls the number of line segments used in representing the curved portions of each polar rectangular solid. It is set to 2 by default. It should be set higher if very few rectangles are used in the grid. Any ParametricPlot3D option may also be used; it will be passed to the graphic on the right.