# 3.3: The Extended Plane

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Consider again inversion about the circle $$C$$ given by $$|z - z_0| = r\text{,}$$ and observe that points close to $$z_0$$ get mapped to points in the plane far away from $$z_0\text{.}$$ In fact, a sequence of points in $$\mathbb{C}$$ whose limit is $$z_0$$ will be inverted to a sequence of points whose magnitudes go to $$\infty\text{.}$$ Conversely, any sequence of points in $$\mathbb{C}$$ having magnitudes marching off to $$\infty$$ will be inverted to a sequence of points whose limit is $$z_0\text{.}$$

With this in mind, we define a new point called the point at infinity, denoted $$\infty.$$ Adjoin this new point to the plane to get the extended plane, denoted as $${\mathbb{C}}^+\text{.}$$ Then, one may extend inversion in the circle $$C$$ to include the points $$z_0$$ and $$\infty\text{.}$$ In particular, inversion of $$\mathbb{C}^+$$ in the circle $$C$$ centered at $$z_0$$ with radius $$r\text{,}$$ $$i_C: \mathbb{C}^+ \to \mathbb{C}^+\text{,}$$ is given by

$i_C(z) = \begin{cases}\dfrac{r^2}{(\overline{z-z_0})} + z_0 & \text{ if $$z \neq z_0,\infty$$; } \\ \infty & \text{ if $$z=z_0$$; } \\ z_0 & \text{ if $$z = \infty$$ } \end{cases}\text{.}$

Viewing inversion as a transformation of the extended plane, we define $$z_0$$ and $$\infty$$ to be symmetric points with respect to the circle of inversion.

The space $${\mathbb{C}}^+$$ will be the canvas on which we do all of our geometry, and it is important to begin to think of $$\infty$$ as “one of the gang,” just another point to consider. All of our translations, dilations, and rotations can be redefined to include the point $$\infty\text{.}$$

So where is $$\infty$$ in $${\mathbb{C}}^+\text{?}$$ You approach $$\infty$$ as you proceed in either direction along any line in the complex plane. More generally, if $$\{z_n\}$$ is a sequence of complex numbers such that $$|z_n| \to \infty$$ as $$n \to \infty\text{,}$$ then we say $$\displaystyle\lim_{n\to\infty} z_n = \infty\text{.}$$ By convention, we assume $$\infty$$ is on every line in the extended plane, and reflection across any line fixes $$\infty\text{.}$$

##### Theorem $$\PageIndex{1}$$

Any general linear transformation extended to the domain $${\mathbb{C}}^+$$ fixes $$\infty\text{.}$$

Proof

If $$T(z) = a z + b$$ where $$a$$ and $$b$$ are complex constants with $$a \neq 0\text{,}$$ then by limit methods from calculus, as $$|z_n| \to \infty\text{,}$$ $$|a z_n + b| \to \infty$$ as well. Thus, $$T(\infty) = \infty\text{.}$$

So, with new domain $${\mathbb{C}}^+\text{,}$$ we modify our fixed point count for the basic transformations:

• The translation $$T_b$$ of $${\mathbb{C}}^+$$ fixes one point ($$\infty$$).
• The rotation about the origin $$R_\theta$$ of $${\mathbb{C}}^+$$ fixes $$2$$ points ($$0$$ and $$\infty$$).
• The dilation $$T(z) = kz$$ of $${\mathbb{C}}^+$$ fixes $$2$$ points, ($$0$$ and $$\infty$$).
• The reflection $$r_L(z)$$ of $${\mathbb{C}}^+$$ about line $$L$$ fixes all points on $$L$$ (which now includes $$\infty$$).
##### Example $$\PageIndex{1}$$: Some Transformations Not Fixing $$\infty$$.

The following function is a transformation of $${\mathbb{C}}^+$$

$T(z) = \dfrac{i+1}{z+2i}\text{,}$

a fact we prove in the next section. For now, we ask where $$T$$ sends $$\infty\text{,}$$ and which point gets sent to $$\infty\text{.}$$

We tackle the second question first. The input that gets sent to $$\infty$$ is the complex number that makes the denominator 0. Thus, $$T(-2i) = \infty.$$

To answer the first question, take your favorite sequence that marches off to $$\infty\text{,}$$ for example, $$1, 2, 3,\ldots\text{.}$$ The image of this sequence, $$T(1), T(2),$$ $$T(3),\ldots$$ consists of complex fractions in which the numerator is constant, but the denominator grows unbounded in magnitude along the horizontal line $$\text{Im}(z) = 2\text{.}$$ Thus, the quotient tends to $$0$$, and $$T(\infty) = 0\text{.}$$

As a second example, you can check that if

$T(z) = \dfrac{iz+(3i+1)}{2iz+1},$

then $$T(\dfrac{i}{2}) = \infty$$ and $$T(\infty) = \dfrac{1}{2}\text{.}$$

We emphasize that the following key results of the previous section extend to $$\mathbb{C}^+$$ as well:

• There exists a unique cline through any three distinct points in $$\mathbb{C}^+\text{.}$$ (If one of the given points in Theorem $$3.2.1$$ is $$\infty\text{,}$$ the unique cline is the line through the other two points.)
• Theorem $$3.2.3$$ applies to all points $$z$$ not on $$C\text{,}$$ including $$z = z_0$$ or $$\infty\text{.}$$
• Inversion about a cline preserves angle magnitudes at all points in $$\mathbb{C}^+$$ (we discuss this below).
• Inversion preserves symmetry points for all points in $$\mathbb{C}^+$$ (Theorem $$3.2.5$$ holds if $$p$$ or $$q$$ is $$\infty$$).
• Theorem $$3.2.7$$ now holds for all clines that do not intersect, including concentric circles. If the circles are concentric, the points symmetric to both of them are $$\infty$$ and the common center.
##### Stereographic Projection

We close this section with a look at stereographic projection. By identifying the extended plane with a sphere, this map offers a very useful way for us to think about the point $$\infty\text{.}$$

##### Definition: Unit $$2$$-Sphere

The unit $$2$$-sphere, denoted $$\mathbb{S}^2\text{,}$$ consists of all the points in $$3$$-space that are one unit from the origin. That is,

$\mathbb{S}^2 = \{(a,b,c) \in \mathbb{R}^3 ~|~ a^2+b^2+c^2=1\}\text{.}$

We will usually refer to the unit $$2$$-sphere as simply “the sphere.” Stereographic projection of the sphere onto the extended plane is defined as follows. Let $$N = (0,0,1)$$ denote the north pole on the sphere. For any point $$P \neq N$$ on the sphere, $$\phi(P)$$ is the point on the ray $$\overrightarrow{NP}$$ that lives in the $$xy$$-plane. See Figure $$3.3.1$$ for the image of a typical point $$P$$ of the sphere. Figure $$\PageIndex{1}$$: Stereographic projection. (Copyright; author via source)

The stereographic projection map $$\phi$$ can be described algebraically. The line through $$N = (0,0,1)$$ and $$P = (a,b,c)$$ has directional vector $$\overrightarrow{NP}=\langle a,b,c-1\rangle\text{,}$$ so the line equation can be expressed as

${\vec r}(t) = \langle 0,0,1\rangle + t\langle a,b,c-1\rangle\text{.}$

This line intersects the $$xy$$-plane when its $$z$$ coordinate is zero. This occurs when $$t = \dfrac{1}{1-c}\text{,}$$ which corresponds to the point $$(\dfrac{a}{1-c}, \dfrac{b}{1-c},0)\text{.}$$

Thus, for a point $$(a,b,c)$$ on the sphere with $$c \neq 1\text{,}$$ stereographic projection $$\phi:\mathbb{S}^2 \to \mathbb{C}^+$$ is given by

$\phi((a,b,c)) = \dfrac{a}{1-c}+\dfrac{b}{1-c}i\text{.}$

Where does $$\phi$$ send the north pole? To $$\infty\text{,}$$ of course. A sequence of points on $$\mathbb{S}^2$$ that approaches $$N$$ will have image points in $$\mathbb{C}$$ with magnitudes that approach $$\infty\text{.}$$

##### Angles at $$\infty$$.

If we think of $$\infty$$ as just another point in $$\mathbb{C}^+\text{,}$$ it makes sense to ask about angles at this point. For instance, any two lines intersect at $$\infty\text{,}$$ and it makes sense to ask about the angle of intersection at $$\infty\text{.}$$ We can be guided in answering this question by stereographic projection, thanks to the following theorem.

##### Theorem $$\PageIndex{2}$$

Stereographic projection preserves angles. That is, if two curves on the surface of the sphere intersect at angle $$\theta\text{,}$$ then their image curves in $$\mathbb{C}^+$$ also intersect at angle $$\theta\text{.}$$

Thus, if two curves in $$\mathbb{C}^+$$ intersect at $$\infty$$ we may define the angle at which they intersect to equal the angle at which their pre-image curves under stereographic projection intersect. The angle at which two parallel lines intersect at $$\infty$$ is 0. Furthermore, if two lines intersect at a finite point $$p$$ as well as at $$\infty\text{,}$$ the angle at which they intersect at $$\infty$$ equals the negative of the angle at which they intersect at $$p\text{.}$$ As a consequence, we may say that inversion about a circle preserves angle magnitudes at all points in $$\mathbb{C}^+\text{.}$$

## Exercises

##### Exercise $$\PageIndex{1}$$

In each case find $$T(\infty)$$ and the input $$z_0$$ such that $$T(z_0) = \infty\text{.}$$

1. $$T(z) = \dfrac{(3 - z)}{(2z + i)}\text{.}$$
2. $$T(z) = \dfrac{(z + 1)}{e^{i\pi/4}}\text{.}$$
3. $$T(z) = \dfrac{(az + b)}{(cz + d)}\text{.}$$
##### Exercise $$\PageIndex{2}$$

Suppose $$D$$ is a circle of Apollonius of $$p$$ and $$q\text{.}$$ Prove that $$p$$ and $$q$$ are symmetric with respect to $$D\text{.}$$

Hint

Recall the circle $$C$$ in the proof of Theorem $$3.2.6$$. Show that $$p$$ and $$q$$ get sent to points that are symmetric with respect to $$i_C(D)\text{.}$$

##### Exercise $$\PageIndex{3}$$

Determine the inverse stereographic projection function $$\phi^{-1}:\mathbb{C}^+ \to \mathbb{S}^2\text{.}$$ In particular, show that for $$z = x + yi \neq \infty\text{,}$$

$\phi^{-1}(x,y) = \bigg(\dfrac{2x}{x^2+y^2+1},\dfrac{2y}{x^2+y^2+1},\dfrac{x^2+y^2-1}{x^2+y^2+1}\bigg).$

This page titled 3.3: The Extended Plane is shared under a CC BY-SA 4.0 license and was authored, remixed, and/or curated by Michael P. Hitchman via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.