Weight:

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 := false
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 := 40
GRAPH3D.STEPSV := 40
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. As for r, since I want shape that is circular in the xz-plane but thin the y's direction, I will use r=1 in the formulas for x and z, but r=0.2 for the formula for y. To help see the coordinates I'm going to start breaking the command between coordinates.

graph3d((rectangular(

cos(U)*sin(V),

0.2*sin(U)*sin(V),

cos(V)))).

Line:

1. The pivot point for the pendulum will be at the point (0,0,4). For now we'll just going to make the line to the point(0,0,0). The equation z=start+(end-start)t will change the z coordinate from a start value when t=0 to the end value when t=1. Since I want z for go from 4 to 0 for now, the equation becomes

z=4+(0-4)t

z=4-4t

  For t, we'll use V/π because

0<V<π

0<V/π<1

  So we now need a second command of

rectangular(0,0,4-4*V/PI)

  But if you try this you won't see the line since

 graph3d.mesh := false

 and the line would have 0 thickness.

2. To give the line thickness, add cos(U)/40 to x and sin(U)/40 to y. So the full graphing command is

graph3d((

rectangular(

cos(U)*sin(V),

0.2*sin(U)*sin(V),

cos(V)),

rectangular(

cos(U)/40,

sin(U)/40,

4-4*V/PI)

)).

  Note , the comma separating the rectangular commands.

Sliding:

1. We need the pendulum to slow down at the ends of the swing and accelerate to the center of the swing. Thus, cos(time) would be good to control the movement. Since we need the swing to go from x=-3 to x=3, we'll add 3*cos(time) to x in both rectangular commands.

graph3d((

rectangular(

cos(U)*sin(V)+3*cos(time),

0.2*sin(U)*sin(V),

cos(V)),

rectangular(

cos(U)/40+3*cos(time),

sin(U)/40,

4-4*V/PI)

)).

Fixing the pivot point:

1. To fix the pivot point we need to have x go from 0 to 3*cos(time) as z goes from 4 to 0. We use the same type of equation.

x=0+(3cos(time)-0)(V/π)

x=3cos(time)(V/π)

  We only make this change for the line.

graph3d((

rectangular(

cos(U)*sin(V)+3*cos(time),

0.2*sin(U)*sin(V),

cos(V)),

rectangular(

cos(U)/40+3*cos(time)*V/PI,

sin(U)/40,

4-4*V/PI)

)).

Swinging:

1. Since a pendulum swings in an arc of a circle in the xz-plane where the center is (0,0,4) and r=distance from (0,0,4) to (0,0,-3)=7. Thus,

x²+(z-4)²=7²

(z-4)²=49-x²

z-4=-(49-x²)^0.5

z=4-(49-x²)^0.5

  Note, we took the negative square root since the weight would be on the bottom of the circle.

  Since,

x=3cos(time)

  Thus,

z=4-(49-(3cos(time))²)^0.5

z=4-(49-9cos²(time))^0.5

  This needs to be added to the z for the weight. For the line,

z=4+(4-(49-9cos²(time))^0.5-4)(V/π)

z=4-(49-9cos²(time))^0.5)(V/π).

  Finally the command is

graph3d((

rectangular(

cos(U)*sin(V)+3*cos(time),

0.2*sin(U)*sin(V),

cos(V)+4-(49-9*cos(time)^2)^0.5),

rectangular(

cos(U)/40+3*cos(time)*V/PI,

sin(U)/40,

4-((49-9*cos(time)^2)^0.5)*V/PI)

)).