Circle is Curve of Second Degree

Theorem

The circle is a curve of degree $2$.

Proof

From Equation of Circle in Cartesian Plane, a circle can be expressed in the form:

$x^2 + y^2 - 2 a x - 2 b y + a^2 + b^2 - R^2 = 0$

where $\tuple {a, b}$ is the center and $R$ is the radius.

This is a quadratic equation in $2$ variables.

Hence the result by definition of degree of curve.

$\blacksquare$

This article, or a section of it, needs explaining. In particular: Sommerville is not rigorous about defining exactly what the degree of a curve is, so the definition needs to wait until such time as I can hunt it down from another source. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by explaining it. To discuss this page in more detail, feel free to use the talk page. When this work has been completed, you may remove this instance of {{Explain}} from the code.

Sources

• 1933: D.M.Y. Sommerville: Analytical Conics (3rd ed.) ... (previous) ... (next): Chapter $\text {III}$. The Circle: $14$. To find the equation of the circle whose centre is $\tuple {\alpha, \beta}$ and radius $r$