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Parabolic Helix

In the 3D space, figuring out the function of a surface using \(x,y,\) and \(z\) can be difficult. Sometimes the surface you want can't pass the vertical line test. This is where parametrics come in. Instead of being a part of a function, \(x,y,\) and \(z\) are the functions themselves. Parametric equations for each coordinate determine the location of a point on your surface. The functions of each coordinate need to have variables of their own, too. The variables that determine the position of each point are \(s\) and \(t\). In parametric equations, \(x, y,\) and \(z\) are only connected to each other through \(s\) and \(t\), so the coordinates don't necessarily have to be related to each other at all. Let's say you want a parabola that extends upward in the 3D space. A good start is to just figure out what the parametric equations look like in the \(xy\) plane. This means \(z=0\) for now. To convert \(y=x^2\) to a parametric equation, you set \(x=t\). Then, you replace every \(x\) with \(t\) so that \(y=t^2\). This makes a parabola in the \(xy\) plane. But what about \(z\)? If we want this parabola to be extended in the \(z\) direction, we can replace \(z\) with \(s\) so that for every value of \(z\), there is a parabola in the \(xy\) plane. That is what a level curve is. It is like a slice of the surface that makes a curve. If we want to know what the curve is at height 5, replace \(z\) with 5 to get \(s=5\). Then replace any \(s\) in the other parametric equations with 5. In this case, we still get the same parabola that we started out with becuase it is just extended in the \(z\) direction and isn't dependent on \(s\). The parametric equations I chose for my surface, though, do rely on \(s\).
 
My surface is a parabola rotating around the origin as it extends upward in the \(z\) direction. Its equations are \[X=t*\cos(s)-(t^2/5)*\sin(s)\]\[Y=t*\sin(s)+(t^2/5)*\cos(s)\]\[Z=2*s\]My parabola is divided by 5 to make the helix thinner and the height is multiplied by 2 to make the helix taller. The helix makes a little more than one rotation. \(s\) ranges from -3.2 to 3.2 and \(t\) ranges from -8 to 8. To explain what all this means, the \(x\) and \(y\) equations are modified to rotate \(s\) radians around the origin. \(z\) is still just \(s\), so as the surface gets taller, the level curve rotates more around the origin. Now, when you replace \(z\) with the height, you can replace \(s\) with that height divided by 2 in the \(x\) and \(y\) equations which ends up rotating the parabola \(s\) radians. You can see in the picture some level curves throughout the surface. At height \(z=6\), the specific equations of \(x\) and \(y\) are \[X=t*\cos(3)-(t^2/5)*\sin(3)\]\[Y=t*\sin(3)+(t^2/5)*\cos(3)\] This gives a parabola rotated about 172 degrees counter-clockwise. When z is negative, the parabola is rotated the same number of degrees clockwise. The reason I picked this surface is because I always thought double helixes were cool looking. Rather than just having straight lines as its level curves, I wanted its curves to be a function of their own and thus, the parabolic helix was created.

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