Problem in understanding the definition of two orthogonal spheres

Question

I was studying about spheres under the topic quadratic surfaces. There, I came accross the term orthogonal spheres. But I couldn't understand what it meant, rigorously. I came acrross a definition in the internet stated as:

Two sphere are said to be orthogonal (or to cut orthogonally) if their tangent planes at a point of intersection are at right angles to each other.

Here, in the above definition I dont seem to get the fact, that since two spheres intersect at more than one point so, will the tangent planes at each of those points of intersection are at right angles to each other? Or will tangent planes at any one of the points of intersection are at right angles to each other will suffice to call two spheres orthogonal? I can't seem to visualize two orthogonal spheres. Can someone provide me with a $3-D$ visualization to help me understand it better?

Answer 1

I can give you a pencil and paper exercise.

Draw a Venn diagram with two circles of equal diameter. There are two points of intersection. Draw the tangent lines to each circle at one of the points of intersection. You will see that the two lines at a single point look like an "X".

As the centers of the two circles get closer together the slopes of the two lines get shallower until they are the same horizontal line when the two circles intersect at every point.

As the centers of the two circles get farther apart, the two tangent lines get steeper until they become the same vertical line when the circles are tangent at a single point.

At some point in between being the same circle and being tangent at a single point, the two tangent lines will be perpendicular. Now rotate the entire figure about the line connecting the two centers and you will have two orthogonal spheres.

The exercise used two circles of equal diameter but now that you see the relationship between the two circles and spheres, you should be able to visualize the case when the spheres have different diameters.

Answer 2

Here is a two dimensional analogue, from wikipedia. The tangents to the blue and red circles at the points of intersection are orthogonal.

This video (which I have not watched) may help.

Answer 3

Suppose that two spheres $(\mathscr{S}_i)$ with respective centers $C_i$ and radii $r_i$ ($i\in\{1,2\}$) meet at some point $M$; and let $(\mathscr{P}_i)$ the two planes respectively tangent to $(\mathscr{S}_i)$ at $M$. Then $(\mathscr{P}_1)$ and $(\mathscr{P}_2)$ are perpendicular iff $\overrightarrow{C_1M}\perp\overrightarrow{C_2M}$, which is in turn equivalent to say that $\triangle(C_1MC_2)$ is rectangle at $M$, or equivalent also to: $\color{red}{r_1^2+r_2^2=C_1C_2^2}$ (by Pythagoras' theorem).

Now observe that the latter relation does not depend on $M$. Put roughly: what happens at some point of $(\mathscr{S}_1)\cap(\mathscr{S}_2)$ happens at any other point of the same set (circle). Another possible explanation. The line $\Delta=(C_1C_2)$ is a symmetry axis for the set $(\mathscr{S}_1)\cup(\mathscr{S}_2)$; if $M,M'\in(\mathscr{S}_1)\cap(\mathscr{S}_2)$, there is some rotation with axis $\Delta$ that maps $M$ to $M'$, and since rotations preserve angles and tangency between surfaces, the angle made by the tangent planes at $M'$ is the same as the angle made by the tangent planes at $M$.