3D Parametric

Post date: Sep 03, 2012 1:2:22 PM

[Update 8/16/2016 - New TI-Nspire file attached at the bottom with the intersection of two spheres. This applies to GPS as described by Jeff in a TI Webinar that aired tonight.

Update: See the discussion and especially the file John Hanna attached on the TI-Nspire google Group.

Update 10/30/2014 - See the attached Tootsie Pop tns file at the bottom - created after Chris B and Amy Oxley's NCTM Indy session. I really enjoyed the modeling sessions #1, 5, 89, 139 ]

3D parametric can be used to do solids of revolution (see the tns file). To rotate f1(x) around the x-axis, let

xp1(t,u)=t

yp1(t,u)=f1(t)*cos(u)

zp1(t,u)=f1(t)*sin(u)

If you want to know a bit more about why this works, think of t=x and u=theta.

So you are getting the two components of the functions mapped to y and z.

If you want a y=f1(x) to be rotated around the y-axis, you'll want to solve for the inverse of y.

xp1(t,u)=inv f1(t)*cos(u)

yp1(t,u)=t

zp1(t,u)=inv f1(t)*sin(u)

I also figured out how to rotate something around x=3. You can see this done on the last page of SolidRev2.tns

For example y=x^2 was rotated about x=3, with xp(t,u)=(3-sqrt(t))*sin(u)+3, yp(t,u)=t, zp(t,u)=(3-sqrt(t))*cos(u).

Adjusting the edit parameters and the range/window was part of the challenge.

The biggest challenge was to do Volume by Cross Sections in 3D. (Enjoy the updated tns file.)

This is shown on a youtube video http://www.youtube.com/watch?v=QNue94Y0xls

3D spheres can now be done with one parametric equation. Now we don't need to do the 'top' part and the 'bottom' part with some 'real' fancy tricks going on to make it look like it is connected (also change the Settings from real to rectangular. See this YouTube video).

The above image is from 0-pi for t and 0 to 2pi for u. These can be changed by tabbing to the [...] “Edit Parameters...”

 Why does this work?  Well, we know a circle can be written as x2 + y2 = radius2, so the equation of a sphere is x2 + y2 + z2 = r2

Think about how to write a circle in 2D parametric. x = r*cos(theta) and y = r*sin(theta). If the r=1, doesn't that remind you of the unit circle. Now, if z is a component of of r, r*cos(t) and the x & y are components of the other component, then perhaps that helps in understanding the above equation for the sphere a bit.

Also enjoy the following 3D parametric Mobius, etc.