Question

Sketch the space curve represented by the intersection of the surfaces. Then represent the curve by a vector-valued function using the given parameter. $$\text { Surfaces } \quad \text { Parameter }$$ $$ 4 x^{2}+4 y^{2}+z^{2}=16, \quad x=z^{2} \quad z=t $$

Short answer

The vector-valued function that represents the space curve at the intersection of the two surfaces is \(r(t) = (t^{2}, \sqrt{4 - t^{2} - t^{4}}, t)\) where \(-\sqrt{2} \leq t \leq \sqrt{2}\).

Step-by-step solution

Understanding the surfaces

The equation \(4x^{2}+4y^{2}+z^{2}=16\) represents an ellipsoid centered at the origin. Meanwhile, the equation \(x=z^{2}\) is a parabolic cylinder. Thus, the resulting space curve will be at the intersection of this ellipsoid and the parabolic cylinder.

Express x, y, z in terms of parameter t

Using the parameter given as \(t=z\), we can express \(x\) and \(y\) in terms of \(t\). Given that \(x=z^{2}\), we can write \(x=t^{2}\). To express \(y\) in terms of \(t\), we need to rearrange the equation of the ellipsoid, resulting in \(y=\sqrt{4 - t^{2} - t^{4}}\).

Vector-valued function

Now we can write down the vector-valued function that represents the intersection of the two surfaces. It is given by \(r(t)=(t^{2}, \sqrt{4 - t^{2} - t^{4}}, t)\), with \(t\) in the interval \(-\sqrt{2} \leq t \leq \sqrt{2}\). This interval is calculated with the consideration of the equation of the cylinder, giving the bounds for \(z\), hence for \(t\).

Key concepts

Ellipsoid

An ellipsoid is a 3-dimensional analogue of an ellipse and is a surface that can be described by an equation of the form \(\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1\). Just like an ellipse, which is defined by two radii, an ellipsoid is characterized by three radii: \(a\), \(b\), and \(c\), which are measurements along the x, y, and z axes, respectively.

These radii determine the length of the ellipsoid along each axis. If all three are equal, the ellipsoid becomes a sphere. In our exercise, the ellipsoid is represented by the equation \(4x^2 + 4y^2 + z^2 = 16\), which can be rewritten as \(\frac{x^2}{2^2} + \frac{y^2}{2^2} + \frac{z^2}{4^2} = 1\) after dividing through by 16, showing that our ellipsoid is stretched twice as far in the z direction as in the x and y directions.

Parabolic Cylinder

A parabolic cylinder is a type of surface in 3-dimensional space. It is formed by translating a parabola along a line that is not on the plane of the parabola, usually perpendicular to the parabola's axis of symmetry. The general equation for such a surface can be represented as \(y - k = a(x - h)^2\) when the parabola opens up or down or \(x - h = a(y - k)^2\) when it opens sideways, where \(h\) and \(k\) are the coordinates of the vertex of the parabola.

In the problem provided, the equation \(x = z^2\) describes a parabolic cylinder with the parabola opening to the right since \(x\) is the dependent variable and no \(y\) is present in the equation. This means that for any value of \(y\), the shape of the curve remains unchanged.

Space Curve

A space curve is a curve that exists in three dimensions. Unlike 2-dimensional curves, which lie flat on a plane, space curves can twist and turn through the x, y, and z dimensions. The intersection of two surfaces in 3D space often forms a space curve. In our exercise, the space curve is the set of all points that are simultaneously on both the ellipsoid and the parabolic cylinder.

To better visualize a space curve, you can imagine it as the path of an insect as it flies through 3D space, sometimes looping, turning, or crossing over itself. The space curve defined by our surfaces is more rigid; it's the exact trail a point on the intersecting line of both surfaces would trace.

Parametric Equations

Parametric equations are a way of defining a geometric object using parameters. In terms of curves, instead of the curve being expressed as \(y\) as a function of \(x\), both \(x\) and \(y\) are expressed as functions of a third variable, usually \(t\). This allows us to represent more complex curves, including those that loop back on themselves, which couldn't be described by a single function in the form \(y = f(x)\).

Parametric equations are particularly useful in representing the motion of an object along a path where \(t\) can denote time. In our original exercise, the parametric equations for the space curve are derived from the intersection of the ellipsoid and parabolic cylinder, resulting in \(r(t) = (t^2, \sqrt{4 - t^2 - t^4}, t)\), where \(t\) represents the parameter \(z\) and binds the values of \(x\), \(y\), and \(z\) together along the curve.