# Finding the equation of a parabola in 3d space

• Infil
In summary, the conversation discusses finding the equation of a parabola in 3D space that goes through three given points and sampling points on the parabola. It is mentioned that the parabola will lie on the plane defined by the three points and the challenge is to transform between the 3D space and the plane's local coordinate system. The suggested solution is to project the points onto the xy-plane, fit a polynomial, and then project it back onto the plane.
Infil
Hi everyone,

I have three points in 3D space, and I would like to find the equation of a parabola that goes between them. My final goal is to sample about 20-25 points that lie on the parabola between these three points (ie, the user of my program will provide 3 points, then I will draw a "dotted line" version of the parabola between them through more discretized sampling).

I know how to find the parabola that goes through 3 points in 2D space, and I know how to find the equation of the unique plane that runs through these 3 points.

I just don't know how to connect the two pieces (or if there is an easier way to accomplish my above goal).

Any help would be greatly appreciated!

The only 3D "parabola" in the 3D space that goes through 3 points is a plane.

Right, the parabola must lie on the plane defined by those 3 points.

I guess what I'm really asking for is a way to transform between the 3D space and the local coordinate system of a plane Ax + By + Cz + D = 0. This way, I can transform the three 3D points to a local 2D coordinate system, solve my problem there, and then transform any point on the plane back to 3D.

This sounds like it should be easy, but I'm drawing a blank. :)

You need to:
1. project the 3 points onto the "xy-plane"
2. fit the polynomial on the xy-plane
3. project the polynomial from the xy-plane to the (Ax + By + Cz + D = 0)-plane.

## 1. What is a parabola in 3d space?

A parabola in 3d space is a three-dimensional curved shape that resembles a U-shape. It is created by the intersection of a plane and a cone that is parallel to one of its sides. It is a type of conic section and can be described by a quadratic equation in three variables.

## 2. How do you find the equation of a parabola in 3d space?

To find the equation of a parabola in 3d space, you need to know the coordinates of three points on the parabola. These points can be used to create a system of equations, which can then be solved to find the coefficients of the quadratic equation. Alternatively, if you know the focus and directrix of the parabola, you can use the standard formula for a parabola in 3d space: (x-h)^2 = 4p(y-k), where (h,k) is the coordinates of the focus and p is the distance from the focus to the directrix.

## 3. What is the significance of the vertex in a parabola in 3d space?

The vertex is the point where the parabola changes direction in 3d space. It is the highest or lowest point on the parabola and is located at the intersection of the parabola and its axis of symmetry. The coordinates of the vertex are important in determining the shape and orientation of the parabola.

## 4. Can a parabola in 3d space have a vertical or horizontal orientation?

Yes, a parabola in 3d space can have a vertical or horizontal orientation, depending on the direction of its axis of symmetry. If the axis of symmetry is parallel to the z-axis, the parabola will have a vertical orientation. If the axis of symmetry is parallel to the x-axis or y-axis, the parabola will have a horizontal orientation.

## 5. How is a parabola in 3d space different from a parabola in 2d space?

A parabola in 3d space is a three-dimensional shape, while a parabola in 2d space is a two-dimensional shape. This means that a parabola in 3d space has a third variable, z, in addition to the x and y variables. The equation of a parabola in 2d space only has two variables, and the parabola is represented by a curve on a flat surface. In contrast, a parabola in 3d space is represented by a curved surface in three-dimensional space.

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