Equation for a tilted parabola in 3D?

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SUMMARY

The discussion centers on deriving the equation for a tilted parabola in 3D, specifically one that is rotated around a line connecting the points (1,0) and (0,1) in the xy-plane. The user seeks to understand how to extract a 3D curve from a standard parabola while maintaining fixed intersection points with the x and y axes. It is clarified that rotating a parabola around a non-axis line does not yield a paraboloid, emphasizing the need for precise mathematical definitions and transformations in 3D space.

PREREQUISITES
  • Understanding of 3D coordinate systems
  • Familiarity with the properties of parabolas
  • Knowledge of geometric transformations, specifically rotation
  • Basic concepts of conic sections and their equations
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  • Study the mathematical principles of 3D transformations
  • Learn about the equations of parabolas in 3D space
  • Explore the concept of conic sections and their applications in geometry
  • Investigate the properties of solids of revolution and their equations
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Mathematicians, physics students, and anyone interested in advanced geometry, particularly those working with 3D transformations and conic sections.

2sin54
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So I was doodling around and came up with a problem of finding an equation for a tilted parabola. Basically, a parabola in xy plane which was rotated around a line that goes through points (1,0) and (0,1), making the curve a 3D one. I realize that this curve is a subset of points that define a paraboloid but I have no clue how to extract one from the other. Any help? Here's a picture for visual guidance http://i.imgur.com/6Esnhkh.png
 
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When you say "a parabola in the xy-plane" which parabola do you mean? If you rotate a parabola around any other line but its axis, the result is NOT a paraboloid.
 
HallsofIvy said:
When you say "a parabola in the xy-plane" which parabola do you mean? If you rotate a parabola around any other line but its axis, the result is NOT a paraboloid.
Sorry for being unclear. I do not mean a solid of revolution. I mean taking a parabola which lies on the xy plane and rotating it (moving all of its points) by some amount of degrees around an axis (in my case a line which goes through points (1,0) and (0,1). If you look at the picture it represents a rotated parabola which "had" its points of intersection with x and y axes fixed.
 

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