# I need a geometric math formula solved

Say I have

- a sphere with a center and a radius

- the center of the sphere is moving along a line

- So it has the position x1 at time 0 and the position x2 at the time t2

- So x(t)=(1-t)*x1 + t*x2

And I have

- a plane

- that is defined by a normal vector n1 and a point on the plane p1 at time 0

- the plane now rotates/moves into a new position p2 and orientation n2 (given as a normal vector) at time t2

- there is a formula for the rotation called Rodriguez formula which quite nicely interpolates the rotational motion ([login to view URL])

- the plane's translation is p(t)=(1-t)*p1 + t*p2

- I am looking for a formula for the time between 0 and t2, when the plane and the sphere collide (All the positions and vectors are in 3d space)?

I have a Maple file and a Python(sympy) file, that might help you. I tried to solve it myself, but calling the solve function doesn't seem to finish.

-----------------

This is the intersection formula for the plane and the sphere:

Let's say that specific aspects of this situation are time dependent:

x(t): the position of the sphere center at time t

p(t): the position of one point on the plane at time t

n(t): the orientation of the plane at time t

Then we can write the plane as an equation:

Y: any point on the plane

the normal vector is perpendicular to the vector between two points on the plane

( Y - p(t) ) · n(t) = 0

this also applies to the unit normal vector

( Y - p(t) ) · n(t) / |n(t)|= 0

In this formula, you can plug in a point L which is not on the plane, and in return, get the distance to this point instead of 0:

( L - p(t) ) · n(t) / |n(t)|= distance between L and plane

The point L is the position of the sphere center

( x(t) - p(t) ) · n(t) / |n(t)|= distance between sphere center and plane

The distance should be the radius of the sphere

( x(t) - p(t) ) · n(t) / |n(t)|= R

Simplify this a little

( x(t) - p(t) ) · n(t) = R · |n(t)|

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