Topology

A parametric curve, as well as the region it encloses can be categorised into different types of curves based on their topology.

Curve §

Simple §

Simple Curve

A parametric curve, $C$, is simple if it does not intersect itself between any of its endpoints. Mathematically, this means no point $(x,y)$ on the curve has multiple $t$ values. In terms of the parametric equation $\vec{c}(t)$, this means $\vec{c}$ is a injective, except possibly at its endpoints (when the curve is a loop).

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#todo https://en.wikipedia.org/wiki/Jordan_curve_theorem

Closed/Open Curve §

Closed/Open Curve

A parametric curve $C$ is closed if it is a loop. Mathematically, it means that the parametric equation defining the curve, $\vec{c}(t)$ for $t\in [a,b]$ has $\vec{c}(a)=\vec{c}(b)$ (the first and last point is the same). A curve that is not closed is an open curve

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Positive Orientation §

Positively Oriented Curve

A simple, closed curve $C$ is positively oriented if the region $D$ enclosed by the curve stays on the left as you follow the curve (move along $t$).

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• Note, $C$ is the boundary of $D$, this is denoted as $C=\partial D$

Note that all positively oriented curves that live on the XY plane (that is, all contours) choose their tangent vector such that they are parallel to the Z-axis.

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Region §

Connected §

Connected Region

A region $D$ on a plane is considered connected if any two points $p_1$ and $p_2$ in $D$ can be joined together by a smooth (possibly piecewise) parametric curve such that the curve is entirely in $D$. More formally, for every pair of points $(p_1,p_2),p_1,p_2\in D$ there exists a parametric equation $\vec{c}(t)$ with $t\in [a,b]$ such that:

• $\vec{c}(a)=p_1$ and $\vec{c}(b)=p_2$
• $\forall t\in [a,b],\vec{c}(t)\in D$

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Simply Connected §

Simply Connected Region

A connected region $D$ is simply connected if every simple, closed curve $C$ that is entirely in $D$ has its own enclosed region, $\partial C$ as a subset of $D$, $\partial C\subseteq D$

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Solid §

Solid Region

A region, $D$ is solid if it is bound by a closed surface $S$, that is, $S$ ‘wraps’ around $D$.

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Surface §

Boundary Curve §

Boundary Curve

#todo Learn about manifolds. The boundary curve, $C$ of a parametric surface $S$ is the curve that is the ‘edge’ of the surface. I currently don’t actually know what it means, but hopefully I can update this later. This is notated as $C=\partial S$

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Open/Closed Surface §

Open/Closed Surface

A parametric surface, $S$ is closed if it has no boundary curve. Else, it is open.

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Induced Positive Orientation §

Induced Positive Orientation

Let $S$ be an oriented surface, oriented with unit normal vector field $\hat{n}$, and $C=\partial S$ be its boundary curve. Then, $S$ induces a positive orientation $\hat{t}$ on $C$ if:

• As you traverse along $C$ (using orientation $\hat{t}$) and use the right hand rule such that your index finger is tangential to $C$ and your middle finger ‘up’ or ‘down’ towards the surface, your thumb is always parallel to $\hat{n}$.
• As you ‘stand’ on the ‘outside’ of $S$ (body parallel to $\hat{n}$), very near to $C$, and walk along a path parallel to $\hat{t}$, the surface $S$ stays on your left while the curve $C$ stays on your right.

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See https://mathinsight.org/definition/positively_oriented_boundary for an alternate definition