# What is the intersection of a plane and a sphere? Is it necessarily a circle? Can it be an ellipse?

(Exer 3.27) Consider the plane $H$ determined by the equation $x + y -z = 0$. What is a unit normal vector to $H$? Compute the image of $E:=H\cap \mathbb S^{2}$ under the stereographic projection $\Phi$.

Asked here and I answered agreeing with OP that the image is the circle $T: = \{|z-(1+i)|^2 = 3\}$, but now I ask:

Q1. Is $(2)$ below wrong?

$$\forall \ \text{planes} \ J, \exists \text{circle} \ C \in \mathbb S^2 : C = J \cap \mathbb S^2 \tag{2}$$

To clarify, compare that (2) is different from (1) below. We know that $$\forall \ \text{circles} \ C \in \mathbb S^2, \exists \ \text{plane} \ J : C = J \cap \mathbb S^2. \tag{1}$$

Q2. Do I go wrong in attempting to disprove (2) as follows?

I think $(2)$ contradicts Exer 3.27: I observe that $E$, whose image is the circle $T$, is both

If $(2)$ is true, then for $J=H, \exists C \in \mathbb S^2: C = H \cap \mathbb S^2 =: E ↯$.

Q3. How is Exer 3.27 consistent with the succeeding Exer 3.28?

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(Exer 3.28) Prove that every circle in $\hat{\mathbb C}$ is the image of some circle in $\mathbb S^2$ under stereographic projection $\Phi$.

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I rephrase: $$\forall \ R \in \hat{\mathbb C}, \exists C \in \mathbb S^2 : \Phi(C)=R$$

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Now consider $T$, the circle in Exer 3.27 s.t. $\Phi(E)=T$.

It seems that by Exer 3.28, for $R= T, \exists C \in \mathbb S^2 : \Phi(C)=T$.

$$\therefore, \Phi(E) = \Phi(C) \ \text{while} \ E \ne C \ \text{, but} \ \Phi \ \text{is bijective ↯ ?}$$

As already verified, all intersections of a circle and plane are circles either geodesic/great or small. I avoid symbols in the answer here.

If 2-parameter surface normal coincides with 1-parameter circle normal then the intersection is a great circle. The intersection is a geodesic.

From differential geometry standpoint $\psi$ is between arc and meridian. Cylindrical coordinates $(r,z)$ Clairaut constant derivative $\dfrac{d(r \sin \psi)}{dz}=0$ forms with respect to an arbitrary North-South polar axis.

If the 2-parameter surface normal does not coincide with 1-parameter circle normal, but makes an angle $\gamma$ of relative latitude then the intersection is a small circle. Intersection is like a parallel circle but not an equator.

Clairaut constant derivative $\dfrac{d(r \sin \psi)}{dz}$ is a constant $=\tan \gamma$ forms with respect to an arbitrary North-South polar axis.

In stereographic projection all non-equatorial sections of a plane passing through North pole of the sphere at angle $\gamma$ to North-South polar plane are small circles.

• Narasimham, is $E$ indeed an ellipse? – BCLC Aug 9 '18 at 5:23
• Narasimham, I posted an answer. How is it please? – BCLC Aug 9 '18 at 6:13

(Q1) I suspect some relationship is being illustrated. The 1st 2 coordinates in $[1,1,-1]$ are the centre of the circle $\Phi(E)$ while the norm of $[1,1,-1]$ s the radius of $\Phi(E)$.

(Q2) Yes if the planes intersect $\mathbb S^2$ or $\emptyset$ is considered a circle in $\mathbb S^2$.

(Q3), (Q4) E is not an ellipse. It is a circle in 3D (see here too) parametrised as (WA)

$$\begin{bmatrix} x_1(t)\\ x_2(t)\\ x_3(t) \end{bmatrix} = \begin{bmatrix} \sqrt{\frac 2 3} \cos[t]\\ -\sqrt{\frac 2 4} \sin[t] - \sqrt{\frac 2 {12}} \cos[t]\\ -\sqrt{\frac 2 4} \sin[t] + \sqrt{\frac 2 {12}} \cos[t] \end{bmatrix} = \begin{bmatrix} \sqrt{\frac 1 3}\\ -\sqrt{\frac 1 {12}}\\ \sqrt{\frac 1 {12}} \end{bmatrix}\sqrt{2}\cos(t)+ \begin{bmatrix} 0\\ -\sqrt{\frac 1 {4}}\\ -\sqrt{\frac 1 {4}} \end{bmatrix}\sqrt{2}\sin(t)$$

Observe that while $x^2+y^2+xy=\frac12$ is an ellipse, $x^2+y^2+xy=\frac12 \wedge z= x+y$ is a circle.

• I guess your major confusion was WA saying ellipse on the intersection. When you set $z=x+y$ into $x^2+y^2+z^2=1$ you get indeed an ellipse, but this is the projection of the intersection onto $xy$-plane, i.e. all the coordinates $(x,y)$ that satisfy both equations. – A.Γ. Aug 9 '18 at 9:16
• @A.Γ. I was thinking projection but unable to express precisely. Thanks! Any idea for (Q1) please? – BCLC Aug 9 '18 at 9:18
• To see that all intersections are circles: 1. Intersection of a plane $z=\text{const}$ and the sphere is a circle (or empty set). 2. Any plane can be rotated to become $z=\text{const}$, and sphere is rotation invariant. – A.Γ. Aug 9 '18 at 9:18
• Is the image of $\Phi$ the plane $z=0$ or $z=-1$ (there are different conventions)? – A.Γ. Aug 9 '18 at 10:17
• I do not see how knowing the normal may help finding the image. The exercise is pretty straightforward without the normal. – A.Γ. Aug 9 '18 at 12:36