MA 391 Composition and Communication

Parametric Surfaces & Level Curves : Elevations of Horsetooth Rock Parametric surfaces are surfaces that exist in \( ℝ^3 \), or in 3D, which is defined by two parameters in \( ℝ^2 \). Thus we have a 3D surface that is given by a function of two parameters, usually \(x\) and \(y\), that gives us a \(z\)—typically thought of as the vertical direction, or the one that "comes towards you." This can be written in a somewhat familiar form(s) as: \[ f(x,y)=z, \\ f(x,z)=y, \\ f(y,z)=x.\] Now that we have established the basics of what a parametric equation is we can move on to a discussion of level curves. A level curve of a two variable function \(f\) is a curve of the form \( f(x,y)=k\), where \(k\) is a constant in the range of \(f\). So, what we are really doing is setting \(k\)—which is a coordinate of a point in relation to the \(z\)-axis—to a constant "height" such that we can draw a curve on our surface that es...

Parameterized surfaces and level curves Let's consider a continuous surface \(z=f(x, y)\) defined for points \((x, y)\) in a domain \(D\) in the \(x-y\) plane, we can use level curves to visualize the function of two variables \(f(x, y)\) without leaving the plane. The level curves can be regarded as an implicit description of a surface. For a given value of \(z=k\) in the value domain of \(z=f(x, y)\), the corresponding level curve is simply the cross section of the graph of \(z=f(x,y)\) and the plane \(z=k\). The equation of the corresponding level curve is defined as follows: \[f(x, y) -k =0\] We can select a series of suitable values of k in the value domain of \(f(x,y)\) for the equation of the level curve, then obtain a series of level curves to describe the surface. The surface to be analyzed The function of parameterized surface that we chose is defined as \[z=f(x,y)\] \[=-2x^2+2y^2 +(x^2+y^2)^2\] \[-1.5 \le x \le 1.5, -0.6 \le y \le 0.6\] ...