Spheres, lines and planes

What is it about?

A sphere $K$ with centre $M$ and radius $r$ consists of all points $X$ at distance $r$ from $M$ :

K:\ (\vec x - \vec m)^2 = r^2 \qquad\Longleftrightarrow\qquad (x_1 - m_1)^2 + (x_2 - m_2)^2 + (x_3 - m_3)^2 = r^2

Point test: substitute a point and compare with $r^2$ (less: inside, equal: on, greater: outside).

Sphere and plane: compare the distance $d = \frac{|\vec n \cdot \vec m - c|}{|\vec n|}$ of the centre from $E:\ \vec n \cdot \vec x = c$ with $r$ . For $d < r$ the plane cuts a circle with centre $M'$ (foot of the perpendicular from $M$ ) and radius $\rho = \sqrt{r^2 - d^2}$ ; for $d = r$ it touches the sphere.

Sphere and line: substituting $\vec x = \vec a + t\vec u$ gives $At^2 + Bt + C = 0$ . The discriminant $D = B^2 - 4AC$ decides: secant ( $D > 0$ ), tangent ( $D = 0$ ) or a line that misses the sphere ( $D < 0$ , no common point).

Tangent plane at $B$ on the sphere: $(\vec x - \vec b) \cdot (\vec b - \vec m) = 0$ .

Try it

Drag $P$ and watch the point test. Point on K
Choose “Sphere and plane” and tap Slide the plane through the sphere. Load
Put the plane through the centre: a great circle with $\rho = r$ . Great circle
Move a vertical line outwards: secant, tangent, line missing the sphere. Tangent · Line missing the sphere
Drag $B$ over the sphere and watch the tangent plane. Tangent plane

Tasks

Task 1: Circle of intersection

$K$ : $M(1, 2, 3)$ , $r = 3$ . Find the centre and radius of the circle in which $x_3 = 5$ cuts $K$ . Load task

Show solution

$d = 2 < 3$ , $M'(1, 2, 5)$ , $\rho = \sqrt{9 - 4} = \sqrt 5 \approx 2.24$ . Show

Task 2: Line and sphere

Find the common points of $K$ and $\vec x = (1, 2, -5) + t\,(0, 0, 1)$ . Load task

Show solution

$(t - 8)^2 = 9$ , i.e. $t^2 - 16t + 55 = 0$ , $D = 36$ , $t = 5$ or $t = 11$ : $S_1(1, 2, 0)$ , $S_2(1, 2, 6)$ . Show

Task 3: Tangent plane

Find the tangent plane to $K$ at $B(3, 4, 4)$ . Load task

Show solution

$\overrightarrow{MB} = (2, 2, 1)$ , so $2x_1 + 2x_2 + x_3 = 6 + 8 + 4 = 18$ . Show

Ideas for the classroom

The equation of a sphere is the distance formula $|\overrightarrow{MX}| = r$ – just like the equation of a circle in the plane.
Before sliding the plane, students can predict how $\rho$ depends on $d$ .
Common stumbling blocks: wrong signs in $(x_k - m_k)^2$ ; comparing with $r$ instead of $r^2$ ; using $r - d$ instead of $\sqrt{r^2 - d^2}$ ; forgetting to divide by $|\vec n|$ ; using $\vec b$ instead of $\vec b - \vec m$ as the normal vector.