How to define a parabola in 3d coordinate system.

In summary, the conversation discusses using a graphing application, Autograph, to model a dome-shaped roof on a building. The speaker needs to define parabolic shapes in a 3D system and is unsure of the basic parabolic function to use. They have three sets of coordinates that need to be on the parabola and suggest using the parameterization P(t) = [t,0,at^2] and rotating it around the x,y, and z axes. This can be done through matrix multiplication and translation, resulting in the general parabola G(t) = R_z(w)R_y(u)R_x(v)P(t)+r.
  • #1
hangainlover
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Currently I am using a graphying application called "Autograph" and modeling a building with a dome shaped roof on top. I need to define parabolic shapes in 3d system.
But i can't do it ( my math knowledge is pretty elementary)
What would be the basic parabolic function in 3d that i can base my model on?
I have three sets of coordinates that have to be on the parabola.
 
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  • #2
in the x y z system, i want this parabolic functionto go through (-75, 36, 0) (0,36, 36) (75,-36,0)
 
  • #3
If we start out with [tex]z=ax^2[/tex] in the xz-plane, we can translate and rotate it to the parabola in 3d. We can obviously get all 3-dimensional parabolas by varying a and doing rotations and translations.

The parameterization is the following: [tex]P(t) = [t,0,at^2][/tex].

http://en.wikipedia.org/wiki/Rotation_matrix#Three_dimensions

Now rotate it around the x,y and z axis by angles v,u and w respectively. We do this by matrix multiplication, so we must multiply [tex]P(t)[/tex] by [tex]R_z(w)R_y(u)R_x(v)[/tex]. After that we perform a translation by the arbitrary vector r.

So the general parabola is [tex]G(t) = R_z(w)R_y(u)R_x(v)P(t)+r[/tex]. You can find this on vector-form by multiplying out the matrices.
 

1. What is a parabola in a 3D coordinate system?

A parabola in a 3D coordinate system is a three-dimensional curve that is symmetric about a line, known as the axis of symmetry. It is formed by the intersection of a plane with a right circular cone, where the plane is parallel to one of the sides of the cone.

2. How do you define a parabola in a 3D coordinate system?

A parabola in a 3D coordinate system can be defined by its equation, which is in the form of y = ax^2 + bx + c, where a, b, and c are constants. Alternatively, it can also be defined by its focus and directrix, which are two fixed points and a fixed line respectively.

3. What are the key features of a parabola in a 3D coordinate system?

The key features of a parabola in a 3D coordinate system include its vertex, focus, directrix, axis of symmetry, and the shape of the curve. The vertex is the point where the parabola makes a sharp turn, the focus is a fixed point that lies on the axis of symmetry, and the directrix is a fixed line perpendicular to the axis of symmetry. The shape of the curve can either be concave or convex, depending on the value of the constant a in the equation.

4. How do you graph a parabola in a 3D coordinate system?

To graph a parabola in a 3D coordinate system, you can plot several points that satisfy the equation of the parabola or use the focus and directrix to draw the curve. You can also use software programs such as GeoGebra or Desmos to graph the parabola accurately.

5. What are some real-life applications of parabolas in a 3D coordinate system?

Parabolas in a 3D coordinate system are commonly used in physics, engineering, and architecture. For example, the path of a projectile can be represented by a parabola in a 3D coordinate system. In architecture, parabolas are used in the design of arches and bridges to distribute weight evenly. They are also used in satellite dishes to reflect and focus incoming signals to a specific point.

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