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Parametric equations are a way to represent a curve by using a parameter, often denoted as \( t \). Instead of a single equation relating \( x \) and \( y \) in the plane or \( x \), \( y \), and \( z \) in space, parametric equations use separate equations for each coordinate in terms of \( t \). This approach allows us to describe more complex curves.

• The equation \( x = f(t) \) provides the horizontal position.
• The equation \( y = g(t) \) gives the vertical position.
• For 3D, \( z = h(t) \) describes the depth or height.

By changing \( t \), we move along the curve, much like drawing with a pen while the paper moves underneath. The choice of \( t \) can determine how the curve is traced in terms of speed and direction. Parametric equations are powerful for modeling situations where the relationship between variables is not easily expressed by a single Cartesian equation.

Space curves are curves that extend into the three-dimensional space. Unlike curves contained within a plane, space curves utilize three coordinate functions: \( x(t) \), \( y(t) \), and \( z(t) \). A vector function like \( \mathbf{r}(t) = \langle x(t), y(t), z(t) \rangle \) defines a space curve.

• These can include familiar shapes, such as lines and circles, as well as more intricate forms like helices.
• A space curve may exhibit twisting or bending that isn't possible in just two dimensions.

The given vector function \( \mathbf{r}(t) = \langle \sqrt{2} \sin t, \sqrt{2} \sin t, 2 \cos t \rangle \) traces a space curve in which the \( x \) and \( y \) components are identical, and shifts in \( z \) are governed by the cosine component. Understanding space curves opens up possibilities for analyzing trajectories and movements in 3D models.

An ellipse in 3D can lie in any plane or be part of a more complex surface. Typically, we express an ellipse using its standard form in one of the coordinate planes. The equation \( \left(\frac{x}{a}\right)^2 + \left(\frac{y}{b}\right)^2 = 1 \) is transformed into 3D depending on the situation.

• In our solved example, the ellipse equation emerges in the \( xz \)-plane: \( \left(\frac{x}{\sqrt{2}}\right)^2 + \left(\frac{z}{2}\right)^2 = 1 \).
• This represents an elliptical shape as seen from the side, projected into the \( xz \)-plane.
• With \( x = y \), the ellipse also projects identically into the \( yz \)-plane.

This illustrates a quarter of an elliptical path as the parameter \( t \) moves from \( 0 \) to \( \pi/2 \). The curve lies on the line \( y = x \) while generating perpendicular elliptical paths on the other coordinate planes, offering a richer spatial perspective than a 2D ellipse.