Spiral with Radius Modulated by Sine:
At the right, you will see a spiral along the z-axis, which has had its radius modulated by a sine term. It was produced in Maple using the following commands:x:=rho*cos(t); y:=rho*sin(t); z:=t/2/Pi/10; rho:=1+1/2*sin(t/10); with(plots): spacecurve([x,y,z],t=0..40*Pi,numpoints=1000, axes=normal,orientation=[45,75],thickness=2);
Spiral with Radius Exponentially Decaying:
The next picture on the right is a spiral along the z-axis, with its radius exponentially decaying as one moves up the z-axis. It was produced in Maple using the following commands:x:=rho*cos(t); y:=rho*sin(t); z:=t/2/Pi/10; rho:=exp(-t/10/Pi); with(plots): spacecurve([x,y,z],t=0..40*Pi,numpoints=1000, axes=normal,orientation=[45,75],thickness=2);
Toroidal Spiral:
This is a spacecurve which spirals around a circle centred on the x-y plane. I call it a toroidal spiral because the spiral is actually on the surface of a torus. It was produced in Maple using the following commmands:x:=(1+1/2*cos(12*t))*cos(t); y:=(1+1/2*cos(12*t))*sin(t); z:=1/2*sin(12*t); with(plots): spacecurve([x,y,z],t=0..2*Pi,numpoints=1000, axes=normal,thickness=2,scaling=CONSTRAINED);
Spiral Around a Spiral:
This started off as a basic spiral of radius 1, oriented along the z-axis. I then decided to make the spacecurve spiral with radius 1/10 around the first spiral. I did this in Maple with the following commands:x:=(1+1/10*cos(12*t))*cos(t); y:=(1+1/10*cos(12*t))*sin(t); z:=t/4/Pi + 1/10*sin(12*t); with(plots): spacecurve([x,y,z],t=0..8*Pi,numpoints=1000, axes=normal,scaling=CONSTRAINED,thickness=2, orientation=[45,75]);
Your Turn!
Your mission is to explore spacecurves using Maple, and try to come up with something interesting. Please print the picture of your spacecurve, along with any Maple commands used to generate it, and attach it to your assignment. I will put the most interesting ones on class web pages.If you would like to explore the commands used in the above examples, I have placed them all into a Maple Worksheet, which you may download using the link at the right.
(Containing the above examples)