Question

A surface \(\mathcal{S}\) in space is described that cannot be defined in terms of a function \(z=f(x, y)\). Give a parametrization of \(\mathcal{S}\). \(\mathcal{S}\) is the elliptic cone \(y^{2}=x^{2}+\frac{z^{2}}{16},\) for \(-1 \leq y \leq 5\).

Short answer

Parametrization: \( \vec{r}(t, y) = (y \cos(t), y, 4y \sin(t)) \).

Step-by-step solution

Understanding the Surface

The given surface is an elliptic cone. The equation of the surface is given by \( y^{2} = x^{2} + \frac{z^{2}}{16} \). The bounds for \( y \) are from \( -1 \) to \( 5 \). This surface cannot be directly described by a function \( z = f(x, y) \) as \( z \) depends on both \( x \) and \( y \) non-linearly.

Choosing Parameters

For parametrization, we choose two parameters: \( t \) for the circular component in the \( xz \)-plane and \( y \) for the linear elevation. Let \( t \) range from \( 0 \) to \( 2\pi \) to cover the circular part and \( y \) range from \(-1\) to \(5\) to cover the height of the cone.

Expressing x, y, and z in Terms of Parameters

Given the equation \( y^2 = x^2 + \frac{z^2}{16} \), one suitable parametrization is: \( x = y \cos(t) \), \( z = 4y \sin(t) \), and \( y = y \). This addresses the elliptical cross-section by scaling \( z \) by 4.

Writing the Parametrization

The parametrization of the surface is given by the vector function: \( \vec{r}(t, y) = (y \cos(t), y, 4y \sin(t)) \) where \( 0 \leq t \leq 2\pi \) and \( -1 \leq y \leq 5 \). This captures the elliptical nature of the surface in the \(xz\)-plane as well as the variation of \( y \).

Key concepts

Elliptic Cone

An elliptic cone is a fascinating type of surface in space that resembles a conical shape with an elliptical cross-section. It's defined mathematically by the equation \( y^2 = x^2 + \frac{z^2}{16} \). Here, the elliptic nature is due to the presence of ellipses in the cross-sections parallel to the base, shown by the coefficients in the equation.
When you see such an equation, notice how it relates \(x\), \(y\), and \(z\), forming a symmetrical cone. The term \( \frac{z^2}{16} \) scales the \(z\) direction, making the cross-sections ellipses instead of circles.
Moreover, bounded by \(-1 \leq y \leq 5\), the elliptic cone cannot be easily expressed as \(z = f(x,y)\) since \(z\) depends non-linearly on both \(x\) and \(y\). This complexity is why an alternative representation, like parametrization, is important.

Vector Function

At the heart of 3D parametrization lies the vector function. It's a powerful tool for describing complicated surfaces like the elliptic cone. Imagine it as a map that assigns a distinct point on the surface for each pair of parameter values.
The given vector function \( \vec{r}(t, y) = (y \cos(t), y, 4y \sin(t)) \) describes the elliptic cone uniquely. Each part of the vector corresponds to \(x\), \(y\), and \(z\)-coordinates respectively, allowing you to trace the entire surface as \(t\) and \(y\) vary.
- \(x = y \cos(t)\): Controls the horizontal component with \(t\) acting like an angle.
- \( y = y \): This straightforwardly represents vertical elevation.
- \( z = 4y \sin(t)\): Scales the motion vertically, emphasizing the elliptical nature in the \(xz\)-plane.
With \(0 \leq t \leq 2\pi\) and \(-1 \leq y \leq 5\), this vector function effectively captures the surface's full span.

Surface in Space

Visualizing a surface in space involves grasping its structure and unique characteristics. In 3D, surfaces can take many shapes and forms, and understanding these can provide deeper insights into their behavior.
The elliptic cone is one such surface. Unlike simple planes or spheres, it provides a blend of linear and elliptical features that are best appreciated through spatial reasoning.
By examining the vector function, you can explore how every point on the surface is a result of specific parameter values. This exploration goes beyond simple graphs; it involves dynamic motion through space, vividly illustrated by the cone.
- The surface captures intricate geometrical relationships.
- It's not confined to a single dimension change.
- Parameters \((t, y)\) allow free navigation across the entire space it occupies.
Thus, surfaces like the elliptic cone can enrich your understanding of geometry, bringing equations to life and inspiring how we perceive mathematical structures in real-world contexts.