Finding the equation of a parabola in 3d space

  • Context: Undergrad 
  • Thread starter Thread starter Infil
  • Start date Start date
  • Tags Tags
    3d Parabola Space
Click For Summary
SUMMARY

This discussion focuses on deriving the equation of a parabola in 3D space that passes through three specified points. The user seeks to sample additional points along this parabola for visualization purposes. The solution involves projecting the three points onto the xy-plane, fitting a polynomial to these points, and then transforming the polynomial back to the original 3D plane defined by the equation Ax + By + Cz + D = 0. This method effectively connects the 2D parabola with its 3D representation.

PREREQUISITES
  • Understanding of 3D coordinate systems
  • Knowledge of polynomial fitting techniques
  • Familiarity with plane equations in 3D space (Ax + By + Cz + D = 0)
  • Experience with geometric transformations and projections
NEXT STEPS
  • Research methods for projecting points onto a plane in 3D space
  • Learn about polynomial regression and fitting in 2D
  • Explore coordinate transformation techniques between 2D and 3D
  • Study algorithms for sampling points along a curve in 3D
USEFUL FOR

This discussion is beneficial for mathematicians, computer graphics developers, and anyone involved in 3D modeling or visualization who needs to represent curves in three-dimensional space accurately.

Infil
Messages
2
Reaction score
0
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!
 
Mathematics news on Phys.org
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.
 

Similar threads

MHB Quadratics: How to determine parabola equation.
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Equation for circle points in 3D
  • · Replies 3 ·
Replies
3
Views
3K
MHB Quadratic Equation Help: Find 3 Points to Determine Answer
  • · Replies 6 ·
Replies
6
Views
2K
Undergrad Is there a formula for this parabola?
  • · Replies 4 ·
Replies
4
Views
3K
MHB Parabola standard form of equation at x = -5
  • · Replies 4 ·
Replies
4
Views
3K
MHB Parabola standard form of equation at x = -1
  • · Replies 2 ·
Replies
2
Views
2K
MHB Normals from a point to a parabola
  • · Replies 7 ·
Replies
7
Views
3K
MHB Finding the equation of the parabola
  • · Replies 4 ·
Replies
4
Views
2K
MHB Finding equation of parabola with 3 points given
  • · Replies 4 ·
Replies
4
Views
6K
High School Cartesian Space vs. Euclidean Space
  • · Replies 10 ·
Replies
10
Views
2K