Curves in \(\mathbb{R}^2\)

Lines

A line passing through the fixed point \(P_0 = (x_0, y_0)\), parallel to the vector \(\vec{v} = \big< a, b \big>\):

\[\begin{align*}
\vec{r}(t) &= t \vec{v} + \vec{r}_0 \\
&= t \big< a, b \big> + \big< x_0, y_0 \big>\\
&= \big< at + x_0, bt + y_0 \big>
\end{align*}\]

Graphs of Functions

If \(y = f(x)\) (\(y\) is a function of \(x\):

\[\vec{r}(t) = \big< t, f(t) \big>\]

If \(x = g(y)\) (\(x\) is a function of \(y\)):

\[\vec{r}(t) = \big< g(t), t \big>\]

Examples:

A parabola \(y = x^2\):

\[\vec{r}(t) = \big< t, t^2 \big>\]

A “sideways” parabola \(x = y^2\):

\[\vec{r}(t) = \big< t^2, t \big>\]

Circles & Ellipses

A circle \(x^2 + y^2 = R^2\), oriented counterclockwise:

\[\vec{r}(t) = \big< R \cos(c t + k), R \sin(c t + k) \big>\]

\(R, c, k\) are constants, \(R, c > 0\).

An ellipse \(\left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1\), oriented counterclockwise:

\[\vec{r}(t) = \big< a \cos(c t + k), b \sin(c t + k) \big>\]

\(a, b, c, k\) are constants, \(a, b, c > 0\)

Examples

The circle \(x^2 + y^2 = 9\), with \(c = 1, k = 0\):

\[\vec{r}(t) = \big< 3 \cos t, 3 \sin t \big>\]

The ellipse \(\frac{x^2}{9} + \frac{y^2}{4} = 1\), with \(c=1, k = 0\):

\[\vec{r}(t) = \big< 3 \cos t, 2 \sin t \big>\]

Curves in \(\mathbb{R}^3\)

Lines

A line passing through the fixed point \(P_0 = (x_0, y_0, z_0)\), parallel to the vector \(\vec{v} = \big< a, b, c \big>\):

\[\begin{align*}
\vec{r}(t) &= t \vec{v} + \vec{r}_0 \\
&= t \big< a, b, c \big> + \big< x_0, y_0, z_0 \big>\\
&= \big< at + x_0, bt + y_0, ct + z_0 \big>
\end{align*}\]

Helices (plural of helix):

A circular helix centered about \(z\)-axis:

\[\vec{r}(t) = \big< R \cos( 2\pi c t), R \sin( 2\pi c t), m t \big>\]

\(R, c, m\) are constants, \(R, c > 0\).

Example:

The helix with \(R = 3\), \(c = 1\) and \(m = 2\):

\[\vec{r}(t) = \big< 3 \cos (2\pi t) , 3 \sin (2\pi t), 2t \big>\]