How to think about parametric surfaces and level curves
Functions with a two-dimensional input and a three-dimensional output can be thought of as drawing a surface in three-dimensional space (source:https://www.youtube.com/watch?v=345SnWfahhY). This surface in \(\mathbb R^{3}\) can be defined by parametric equations with two parameters, \({\displaystyle \mathbf {r} :\mathbb {R} ^{2}\to \mathbb {R} ^{3}}\). These parametric equations are a group of functions of one or more independent variables, or parameters, that define a surface in three dimensions (source:https://en.wikipedia.org/wiki/Parametric_surface). For example, a surface can be represented parametrically by taking a point \((u, v)\) in a 2D space "\(D\)" and plugging this point into \[\vec{r}(u,v)=x(u,v)\vec{i}+y(u,v)\vec{j}+z(u,v)\vec{k}.\] The resulting vectors are the position vectors for the points on the surface "\(S\)". The parametric equations of this surface would then be \[x=x(u,v)\] \[y=y(u,v)\] \[z=z(u,v).\] As you can see, these are just the components of the parameteric representation (source:https://tutorial.math.lamar.edu/Classes/CalcIII/ParametricSurfaces.aspx).
The hardest thing to grasp about this concept is how to visualize the surface. It's not intuitive to look at a function and picture what it looks like in a 3D space. To make it easier, consider the topographical map below:
(Image from:https://maps2anywhere.com/maps/ecuador-road-map.html?SID=d8bd3b1ed8135118f331d7d3d8a20ef1)
Cotopaxi is an active volcano in the Andes I summited this summer, and we used a very similar map to plan our ascent. The paper of this map is flat, but I can say from experience this mountain is TALL! However, by using the lines pictured, we could tell which areas would change elevation the most. The elevation change between lines is constant, so the closer together these lines appear, the larger the elevation change over a shorter distance is. We can do the same thing for functions of two variables; the surface can be layed flat on the \(xy\) plane and the height is represented by level curves. Each level curve represents the points where the height of the function is the same and, again, the change in height between the lines is constant.
Why I chose the surface and parameterization I did
Consider the surface given by \(f(x,y)=ln({|y-x^2|}).\)
This shape is not easy to visualize just from looking at our equation; it's something I haven't seen before and found by messing around with GeoGebra until something looked weird. This way, I think the level curves will be especially helpful.
Also, I chose to give the points of my surface as \((x,y,f(x,y))\) instead of \((x(u,v),y(u,v),z(u,v))\) because parametric equations, while very useful, can often make functions appear more complicated than they are (for example, parameterizations with trig functions). I wanted to limit how intimidated someone learning this topic may feel, and generally functions in the form I chose are more intuitive to us because we've seen them this way our whole lives.
To begin finding the level curves, one must first select an elevation. Let's call it "c". We set the equations \(f(x,y)=ln({|y-x^2|})\) and \(f(x,y)=c\) equal to one another and solve for y. This will give us an equation for our level curves. \[ln({|y-x^2|})=c\] \[e^{ln({|y-x^2|})}=e^c\] \[|y-x^2| = \pm e^c\] \[y=x^2 \pm e^c\] We now know that our level curves are represented by parabolas! That's much easier to think about than natural logs. Set the value of "c" to \(\pm1,\pm2,\) and \(\pm3\) to obtain the following graphs:
1. Positive "c"
2. Negative "c"
Now, we can put everything together from our GeoGebra 3D model and Desmos level curves to see the surface with its level curves.
As height increases, the level curves are farther apart. This means the function is increasing more slowly in the positive y direction as x increases. It's interesting to see that we have an inner and outer parabola at some heights due to the nature of our chosen function. My print is going to be scaled to be roughly 2x2 inches at its base so that it is large enough to see the space between curves when viewed from the top.
Word count: 593
Functions with a two-dimensional input and a three-dimensional output can be thought of as drawing a surface in three-dimensional space (source:https://www.youtube.com/watch?v=345SnWfahhY). This surface in \(\mathbb R^{3}\) can be defined by parametric equations with two parameters, \({\displaystyle \mathbf {r} :\mathbb {R} ^{2}\to \mathbb {R} ^{3}}\). These parametric equations are a group of functions of one or more independent variables, or parameters, that define a surface in three dimensions (source:https://en.wikipedia.org/wiki/Parametric_surface). For example, a surface can be represented parametrically by taking a point \((u, v)\) in a 2D space "\(D\)" and plugging this point into \[\vec{r}(u,v)=x(u,v)\vec{i}+y(u,v)\vec{j}+z(u,v)\vec{k}.\] The resulting vectors are the position vectors for the points on the surface "\(S\)". The parametric equations of this surface would then be \[x=x(u,v)\] \[y=y(u,v)\] \[z=z(u,v).\] As you can see, these are just the components of the parameteric representation (source:https://tutorial.math.lamar.edu/Classes/CalcIII/ParametricSurfaces.aspx).
The hardest thing to grasp about this concept is how to visualize the surface. It's not intuitive to look at a function and picture what it looks like in a 3D space. To make it easier, consider the topographical map below:
Cotopaxi is an active volcano in the Andes I summited this summer, and we used a very similar map to plan our ascent. The paper of this map is flat, but I can say from experience this mountain is TALL! However, by using the lines pictured, we could tell which areas would change elevation the most. The elevation change between lines is constant, so the closer together these lines appear, the larger the elevation change over a shorter distance is. We can do the same thing for functions of two variables; the surface can be layed flat on the \(xy\) plane and the height is represented by level curves. Each level curve represents the points where the height of the function is the same and, again, the change in height between the lines is constant.
Why I chose the surface and parameterization I did
Consider the surface given by \(f(x,y)=ln({|y-x^2|}).\)
To begin finding the level curves, one must first select an elevation. Let's call it "c". We set the equations \(f(x,y)=ln({|y-x^2|})\) and \(f(x,y)=c\) equal to one another and solve for y. This will give us an equation for our level curves. \[ln({|y-x^2|})=c\] \[e^{ln({|y-x^2|})}=e^c\] \[|y-x^2| = \pm e^c\] \[y=x^2 \pm e^c\] We now know that our level curves are represented by parabolas! That's much easier to think about than natural logs. Set the value of "c" to \(\pm1,\pm2,\) and \(\pm3\) to obtain the following graphs:
1. Positive "c"
Word count: 593
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