Quadratic Surfaces

Posted on 2008-03-06 04:30 By samjlee87 (1)

Categories:

How do I ...? with Maple (Student)

Graph the surfaces y= x² + z² and y = 1 - z² on the same axes using the domain |x| < or = 1.2 , |z| < or = 1.2 , and observe the curve of intersection of both surfaces. Show that the projection of this curve onto the xz-plane is an ellipse.

So that's that problem. I am so lost on what to do. I know it has to do something with implicitplot3d.

Also, there is a problem where 2 parametric curves were defined by 2 formulas, and they had "control points" which kind of dictate how they look depending on the value of these "control points." Now, given the equation for the parametric equations, how would I plot a linear line from one point to another. There are 4 points so there should be 3 lines

Ellipse

Comment on 2008-03-06 05:43 from John Fredsted ( Maple Rating 4

566)

You can do something like the following:

y1 := (x,z) -> x^2 + z^2:
y2 := (x,z) -> 1 - z^2:
plot3d(
	[y1(x,z),y2(x,z)],x = -1.2..1.2,z = -1.2..1.2,
	orientation = [40,50],labels = [x,z,y],axes = BOXED
);

The projection to the x-z-plane of the intersection curve of the two surfaces is given by the equation y1(x,z) = y2(x,z) which may be rearranged as x^2 + 2*z^2 = 1 which is the equation of an ellipse with major axis a = 1 along the x-axis, and minor axis b = 1/sqrt(2) along the z-axis, visually consistent with the above plot.

improvement

Comment on 2008-03-06 09:57 from Doug Meade ( Maple Rating 4

721)

I do not "see" the elliptical intersection in this plot. Try this modification, with two more options and a different orientation:

y1 := (x,z) -> x^2 + z^2:
y2 := (x,z) -> 1 - z^2:
plot3d(
	[y1(x,z),y2(x,z)],x = -1.2..1.2,z = -1.2..1.2,
	orientation = [30,15],labels = [x,z,y],axes = BOXED,
        style=surface, color=[pink,cyan]
);

See ?plot3d,option for a complete set of options that you could use.

I hope this is helpful,

Doug

---------------------------------------------------------------------
Douglas B. Meade  <><
Math, USC, Columbia, SC 29208  E-mail: mailto:meade@math.sc.edu       
Phone:  (803) 777-6183         URL:    http://www.math.sc.edu/~meade/

Options

Comment on 2008-03-06 10:09 from John Fredsted ( Maple Rating 4

566)

Oh boy, all those plot options :-).

I guess that it boils down to too little plot experience on my behalf.

The ellipse

Comment on 2008-03-06 10:11 from jaytreiman (

41)

To get the ellipse you can note that you are solving two equations in three unknowns. Eliminating y gives you the equation of the ellipse.

The ellipse

Comment on 2008-03-06 10:23 from Scott03 ( Maple Rating 4

468)

Now that you have seen the two surfaces from the previous plot commands, if you would like to see that curve that is the intersection between those two surfaces you could try the following:

> with(plots, intersectplot);
> intersectplot(x^2-y+z^2 = 0, y+z^2 = 1, x = -1 .. 1, y = -1 .. 1, z = -1 .. 1, axes = boxed);

The above will only work for Maple 11 since the intersectplot is a new command for Maple 11.

Scott