Ball:

1. We'll use the rectangular command in DPGraph. Its syntax is

rectangular(x,y,z)

where x,y,z are the coordinates of the points to plot.

2. Then, we'll use a spherical transformation. Spherical coordinates are:

r=distance from origin

θ=horizontal angle from x-axis.

Ф=angle off z-axis.

   It can be shown that the transformation to rectangular coordinates are

x=r cos(θ) sin(Ф)

y=r sin(θ) sin(Ф)

z=r cos(Ф).

3. In DPGraph we have 2 parameters of U and V. We'll use U as θ, and V as Ф. So U will run from 0 to 2π,and V from 0 to π. Also, we'll graph in a viewing box of -4 to 4 for x, y and z. So our setup commands will be:

graph3d.box := true
graph3d.mesh := true
graph3d.view := front
graph3d.perspective := true
graph3d.resolution := 30
graph3d.highlight := 1
graph3d.shading := 0
graph3d.contrast := .5
graph3d.transparency := 0
graph3d.color := blue
graph3d.minimumx := -4
graph3d.maximumx :=4
graph3d.minimumy := -4
graph3d.maximumy := 4
graph3d.minimumz := -4
graph3d.maximumz := 4
GRAPH3D.STEPSU := 20
GRAPH3D.STEPSV := 20
GRAPH3D.MINIMUMU := 0
GRAPH3D.MAXIMUMU := 2*PI
GRAPH3D.MINIMUMV := 0
GRAPH3D.MAXIMUMV :=PI

4. Now putting the transformations into the rectangular command using U and V for θ and Ф respectively and r=1, we get

graph3d((rectangular(cos(U)*sin(V),sin(U)*sin(V),cos(V)))).

Rotating Ball:

1. To make the ball rotate add the variable time to U. To help see the coordinates I'm going to start breaking the command between coordinates.

graph3d((rectangular(

cos(U+time)*sin(V),

sin(U+time)*sin(V),

cos(V)))).

Sliding Ball:

1. To make the ball side we need to move the center of the sphere from -3 to 3 along the x axis.

2. Sin(time) will go from -1 to 1 in a smooth curve, but it would look better to have the ball bouncing off the sides. So arcsin(sin(time)) would look better since y=arcsin(sin(time)) oscillates like y=sin(time), but moves straight from max to min.

3. arcsin(sin(time)) would go from -π/2 to π/2. To make it go from -3 to 3 multiply by 6/π.

4. Finally since this is a shift in x, add this to the x coordinate. Note, in DPGraph arcsin(x)=asin(x).

graph3d((rectangular(

cos(U+time)*sin(V)+6*asin(sin(time))/pi,

sin(U+time)*sin(V),

cos(V))))

Bouncing Ball:

1. Since falling bodies fall in parabolas, we will want the center of the ball to move on a parabola with vertex at (0,0,3) and passing though (-3,0,-3) and (-3,0,3). The parabola z=3-(2/3)x² will do this.

2. Note the x is given by 6arcsin(sin(time))/π . So

z=3-(2/3)(6arcsin(sin(time))/π)²

3. This gets added to the z coordinate.

graph3d((rectangular(

cos(U+time)*sin(V)+6*asin(sin(time))/pi,

sin(U+time)*sin(V),

cos(V)+3-(2/3)*(6*asin(sin(time))/PI)^2)))

Bouncing Beach Ball:

1. The ball is blue because of the command

graph3d.color := blue

   This can be changed to an expression of x,y,z,U, or V.

2. Since most beach balls have multi-colored panels going around it, change this command to

graph3d.color := U/2/PI

  This will make the color values range from 0 to 1, and cause the to change with the horizontal angle.