Curves and Surfaces in (ℝ^3): Traces & Surfaces

Practice Problems relating to curves and surfaces. Solutions are provided.

Problem 1

Find the trace in the z x plane and the trace in the z y plane for the following function.Then get a three dimensional sketch of the function.

f[x_,y_]=Log[x] + y

Your Solution

Barry's Solution

For the trace in the x z plane, we must have x>0 for the logarithm to be defined. The curves are logarithms, and are shifted vertically by varying k.

list=Table[f[x,k],{k,-2,2,1/2}]
Plot[Evaluate[list],{x,0,2},AxesLabel->{"x","z"}]

For the trace in the y z plane, we must have k>0 for the logarithm to be defined. The curves are straight lines, with slope 1.

list=Table[f[k,y],{k,1/2,10}]
Plot[Evaluate[list],{y,-2,2},AxesLabel->{"y","z"}]

Plot3D[f[x,y],{x,0,2},{y,-2,2},AxesLabel->{"x","y","z"},PlotPoints->40]

Problem 2

Find the trace in the z x plane and the trace in the z y plane for the following function.Then get a three dimensional sketch of the function.

f[x_,y_]=16(x^2/4+y^2/9)

Your Solution

Barry's Solution

For the trace in the x z plane, we get parabolas.

list=Table[f[x,k],{k,-2,2,1/2}]
Plot[Evaluate[list],{x,-2,2},AxesLabel->{"x","z"}]

For the trace in the y z plane, we also get parabolas.

list=Table[f[k,y],{k,-5,5}]
Plot[Evaluate[list],{y,-10,10},AxesLabel->{"y","z"}]

Plot3D[f[x,y],{x,-4,4},{y,-4,4},AxesLabel->{"x","y","z"},PlotPoints->40]

Aside: This surface is called a paraboloid since both the traces are parabolas.

Problem 3

Find the trace in the z x plane and the trace in the z y plane for the following function.Then get a three dimensional sketch of the function, and describe the surface.

f[x_,y_]=Sqrt[16-x^2-y^2]

Your Solution

Barry's Solution

For the trace in the x z plane, we get the top half of circles

list=Table[f[x,k],{k,-2,2,1/2}]
Plot[Evaluate[list],{x,-5,5},AxesLabel->{"x","z"}]

For the trace in the y z plane, we also get the top half of circles

list=Table[f[k,y],{k,-5,5}]
Plot[Evaluate[list],{y,-5,5},AxesLabel->{"y","z"}]

Plot3D[f[x,y],{x,-4,4},{y,-4,4},AxesLabel->{"x","y","z"},PlotPoints->40]

Aside: This surface is the top half of a sphere. It is easier to plot the sphere as a parametric function than as an explicit function, just as it is easier to plot a circle as a parametric function than as an explicit function.

Problem 4

Find the trace in the z x plane and the trace in the z y plane for the following function.Then get a three dimensional sketch of the function, and describe the surface.

f[x_,y_]=-Sqrt[16-9x^2-4y^2]

Your Solution

Barry's Solution

For the trace in the x z plane, we get the bottom half of circles

list=Table[f[x,k],{k,-2,2,1/2}]
Plot[Evaluate[list],{x,-2,2},AxesLabel->{"x","z"}]

For the trace in the y z plane, we get the top half of ellipses.

list=Table[f[k,y],{k,-5,5}]
Plot[Evaluate[list],{y,-2,2},AxesLabel->{"y","z"}]

Plot3D[f[x,y],{x,-2,2},{y,-2,2},AxesLabel->{"x","y","z"},PlotPoints->40]

Aside: This surface is the bottom half of what is known as an ellipsoid. An ellipsoid has all traces as ellipses (remember, a circle is just a special ellipse!).

Problem 5

Find the trace in the z x plane and the trace in the z y plane for the following function.Then get a three dimensional sketch of the function, and describe the surface.

f[x_,y_]=Cos[x]

Your Solution

Barry's Solution

For the trace in the x z plane, we get a cosine function.

list=Table[f[x,k],{k,-2,2,1/2}]
Plot[Evaluate[list],{x,-7,7},AxesLabel->{"x","z"}]

For the trace in the y z plane, we get constant functions.

list=Table[f[k,y],{k,-5,5}]
Plot[Evaluate[list],{y,-2,2},AxesLabel->{"y","z"}]

Plot3D[f[x,y],{x,-7,7},{y,-2,2},AxesLabel->{"x","y","z"},PlotPoints->40]