Curves in differential geometry
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This page covers mathematical example of curves in differential geometry.
Constant curve
Given a point p0 in R3 and a subinterval I of the real line,
- <math> \mathbf{\gamma}:t \mapsto \mathbf{p_0} = \begin{pmatrix}
x_0\\ y_0\\ z_0\\
\end{pmatrix}\qquad (t \in I)<math>
defines the constant curve, a parametric curve of class C∞. The image of the constant curve is the single point p. The curve is closed and analytic but not simple.
Line
A slightly more complex example is the line. A parametric definition of a line through the points p0 and p1 (p0 ≠ p1 and p0,p1 ∈ R3) is given by
- <math> \mathbf{\gamma}:t \mapsto \mathbf{p_0} + t(\mathbf{p_1} - \mathbf{p_0})= \begin{pmatrix}
x_0 + t (x_1 - x_0)\\ y_0 + t (y_1 - y_0) \\ z_0 + t (z_1 - z_0) \\
\end{pmatrix} \qquad (t \in I) <math>
The image of the curve is a line. Note that
- <math> \mathbf{\gamma}:t \mapsto \mathbf{p_0} + t^3 (\mathbf{p_1} - \mathbf{p_0})= \begin{pmatrix}
x_0 + t^3 (x_1 - x_0)\\ y_0 + t^3 (y_1 - y_0) \\ z_0 + t^3 (z_1 - z_0) \\
\end{pmatrix} \qquad (t \in I) <math>
is a different curve but the image of both curves is the same line.
Helix
Give r, b in R
- <math> \mathbf{\gamma}:t \mapsto \begin{pmatrix}
r \cos (\omega t)\\ r \sin (\omega t)\\ bt\\
\end{pmatrix}\qquad (t \in I)<math>
defines the helix.