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This page covers mathematical example of Curve s in Differential Geometry .


CONSTANT CURVE

Given a point ''p0'' in R3 and a subinterval ''I'' of the real line,

: \mathbf{\gamma}:t \mapsto \mathbf{p_0} = \begin{pmatrix}
x_0\
y_0\
z_0\
\end{pmatrix}\qquad (t \in I)

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 ''p''0 and ''p''1 (''p''0 ≠ ''p''1'' and ''p''0,''p''1'' ∈ '''R'''3) is given by

: \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)

The image of the curve is a line. Note that

: \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)

is a different curve but the image of both curves is the same line.


HELIX

Given ''r'', ω in R

: \mathbf{\gamma}:t \mapsto \begin{pmatrix}
r \cos (\omega t)\
r \sin (\omega t)\
t\
\end{pmatrix}\qquad (t \in I)

defines a Helix circling the ''z''-axis.