2018 05 20 Tangent Plane To Sphere

0:00 I found instructions for the derivation of an equation for a plane at www.songho.ca/math/plane/plane.html.

2:00 The equation is (a,b,c).(x-x1,y-y1,z-z1) = 0 which represents "the dot product of the normal, and a vector in the plane is zero."

5:00 Working out normal vector and point in a plane, for a circle in the xy plane.

6:18 Finding the Mathematica "InfinitePlane" function, that takes three points in the space, and creates a plane.

7:00 generate three points using (x,y,z) = (0,0,z), (0,y,0) and (x,0,0) where x, y, z satisfy

7:50 Typo here. I have z=r Cos[theta] It should be z=r Sin[theta]

8:20 Writing pseudocode, saying that the dot-product of the normal and the vector in the plane is zero.

9:00 What are those somethings?

9:10 Aside. Make the normal have unit length.

10:00 I have build the dot products for each undetermined points in the plane (x,0,0), (0,y,0), (0,0,z) I solve each of these to make the dot product zero. (Typo from 7:50 remains, so answers aren't correct.)

13:00 Plugging the quantities into the InfinitePlane Graphics Primitive, and Manipulating r, theta, phi.

14:00 Due to division by sin[theta]=0 graphics do not show up, correctly, at first.

14:45 We are able to see the error that comes from the typo at 7:50

15:00 I try re-thinking my aside at 9:10, but it has no effect on the issue.

15:40 I discover my typo from 7:50

17:00 Something is still wrong... I have left some 2's in the calculation.

17:45 Now it's fixed, and the tangent plane stays right on the ball (except when we have division by zero)