Parabolas are a type of curve defined by a single algebraic equation. Parabolas can be shifted and scaled vertically and horizontally, but still maintain their characteristic shape.
When parabolas are rotated about any axis, they retain the same shape, but are now oriented in a different direction. A horizontal parabola rotated about the vertical axis is now a vertical parabola.
It is possible to find an equation for the surface obtained by rotating a parabola about any axis. This article will explain how to find this equation. This article will also explain how to find the surface obtained by rotating a horizontal parabola about the Y-axis.
This article is written for people with a basic understanding of algebra and geometry. If you have questions or concerns, you may ask them in the comments section.
Rotate the parabola about the y-axis
Now, let’s consider the case when we rotate the parabola about the y-axis. As mentioned before, this can be thought of as rotating the function Y = X2 around the x-axis.
When we do this, every point on the parabola moves in parallel along the x-axis and is doubled. This is easy to see by looking at a single point on the parabola:
If (x,y) is a point on the parabola then its image is (x,2y).
So, if we start with a line that goes through the coordinate plane with length 2, when we rotate it around the x-axis it will have length 1. And all points on that line will move in parallel to themselves and be doubled.
To find out what shape we get by rotating Y = X2 about the y-axis, we just have to solve for y in our equation for Y=X2 and then replace y with -y.
The surface area of the rotating parabola
A parabola can be thought of as a surface that is obtained by rotating a line about a direct axis. In math terms, this means that a parabola is a set of points obtained by considering any point on the line y = x and shifting it to the right or left by some value x, and then multiplying it by some scalar (non-scalar) value x.
Parabolas can be rotated about any axis. When we rotate a parabola about the y-axis, we get what is called a torus. A torus is similar to a doughnut shape, where if you slice it down the middle, you would get two rings that look like mirrors of each other.
In mathematics, this shape is defined as the set of all points in space that are equidistant from some fixed plane.
The volume of rotating parabola
A related interesting problem is to find the volume of the solid obtained by rotating the parabolic surface about the y-axis. This problem was first solved by Gabriel Lamé in 1844.
The surface is given by
where F(x,y) is a continuous function on (0,1)2 such that F(0,y)=0 and F′(x,y)=0 for all (x,y) in (0,1)2. The rotation angle ρ can be any real number.
What happens if you rotate more?
If you rotate the parabola about the x-axis, you get a line. If you rotate it about the y-axis, you get a circle. If you rotate it about any axis, you get a surface called a sphere.
If you rotate any curve about an arbitrary axis, the curve becomes a surface called a torus. More generally, if the curve is given in parametric form as , where and are constants, then rotating about an axis translates to
where is some constant depending on which axis you rotated by how much.
Parametric equations for spheres can be found here. Parametric equations for toruses can be found here.
What happens if you rotate less?
If you rotate the parabola about the y-axis by an angle less than π/2, then the surface obtained will be a cylinder with a parabolic cross-section.
This is because if you look at the equation for the rotated parabola, X = X0 + Y0 and if Y = 0 then X = X0 + 0 = X0 which means that X becomes equal to X0 which is the length of the axis of the cylinder.
You can see this in the image below where blue is the axis of the cylinder and where there is a red line going through that point that goes through the center of the circle.
Find an equation for a surface obtained by rotating a hyperbola about its axis
A hyperbola has two axes, a and b, where a > b. A parabola has one axis, the x-axis, where x = y. So to rotate a parabola about the x-axis would produce a hyperbola!
The surface obtained by rotating a hyperbola about its axis is also called a skew curve. Like surfaces obtained by rotating a curve about an axis, skew curves have no sharp corners and no boundaries along the curve being rotated.
To find an equation for the surface obtained by rotating a hyperbola about its axis, first write an equation for the hyperbola using variables x and y. Then solve for y in this equation, then replace y with -y to get an equation for the inverse of the hyperbola.
For example, solve this problem: Find an equation for the surface obtained by rotating the hyperbola 4×2 + 9x − 12y2 = 24 about the y-axis.
What are spheres and ellipsoids?
Besides paraboloids, there are two other common surfaces that can be created by rotation. A sphere is a surface obtained by rotating a circle about its diameter. An ellipsoid is a surface obtained by rotating an ellipse about its major or minor axis.
Spheres can be defined in several different ways. One way to define a sphere is as the set of all points in three-dimensional space that are the same distance from a given point, the center of the sphere.
Another definition is the surface of a ball with a given point as its center and all points on the surface corresponding to points in space on the same plane as this center point and at the same distance from it.
An important note is that both of these definitions assume that the given point in question is inside of the sphere, not outside of it.
How are spheres different from ellipsoids?
Spheres are the most simple of the closed surfaces. A sphere is defined as a surface formed by rotating a circle about its diameter.
Unlike an ellipse, which is a special type of curve, a sphere is a surface that is formed by rotating a circle about its diameter. As such, spheres are characterized only by their diameter and not by any additional parameters.
The other closed surfaces can be described as an ellipsoid. An ellipsoid is defined as a quadrilateral where two of the sides are parallel and the length of the other two sides are constants. Like an ellipse, these constants vary depending on which direction you measure them in.
The interesting thing about ellipsoids is that they can be measured in different ways and still be the same shape! One way to measure an ellipsoid is to measure its axes and then calculate its volume and radius based on those measurements.
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