----- > [!proposition] Proposition. ([[circle arc iff normal lines pass through fixed point]]) > For a [[regular curve|regular]] plane curve $\alpha(t)$, we define the **normal line** of $\alpha$ at $t$ as the line passing through $\alpha(t)$ and [[orthogonal|perpindicular]] to the [[velocity vector of a parameterized curve|tangent vector]] at $t$. > \ > The curve $\alpha$ is part of a circle if and only if all of its normal lines pass through a fixed point. > [!proof]- Proof. ([[circle arc iff normal lines pass through fixed point]]) > [[circles have constant curvature]] coupled with the [[fundamental theorem of the local theory of parameterized curves]] tells us that $\alpha(t)=(x(t),y(t))$ is part of a circle of radius $R$ centered at $(a,b)\in \mathbb{R}^{2}$ iff we have $(x(t)-a)^{2} + (y(t)-b)^{2}=R^{2}$and iff its curvature $\kappa(t)$ is a constant $\frac{1}{R}$. [[derivative|Differentiating]] both sides and dividing by 2 we see that this is equivalent to requiring $(x-a)x'(t) + (y-b)y'(t)=0 \text{ for all }t.$ But we can recognize this equation as $(x(t)-a, y(t)-b) \cdot (x'(t), y'(t))=0.$ So the point $(a,b)$ is contained in the normal line at $t$, and since $t$ was arbitrary, we conclude that $(a,b)$ is contained in *all* normal lines of the curve. ----- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```