What is the formula of ellipse in 3D space

[fatjjx]

the regular ellipse formula in 2D is x^2/a^2 + y^2/b^2 = 1. but how can it be transformed into a 3D formula including the parameter of z?

thank you!

 

[symbolipoint]

Try thinking in the other direction:
Would an ellipse in ONE dimension be (x^2)/(a^2)=1 ?

Do you accept that an ellipse in TWO dimensions is (x^2)/(a^2)+(y^2)/(b^2)=1 ?

Now how would you use three dimensions?

 

[fatjjx]

you are right! but what is the 3D formula of a ellipse??
that is the real problem. thank you! :)

Try thinking in the other direction:
Would an ellipse in ONE dimension be (x^2)/(a^2)=1 ?

Do you accept that an ellipse in TWO dimensions is (x^2)/(a^2)+(y^2)/(b^2)=1 ?

Now how would you use three dimensions?

 

[ron_jay]

what about (x^2/a^2) + (y^2/b^2) + (z^2/c^2)=1 ? (intuitive)

However, the equation of a circle in 3D is always defined by the equation of a sphere and the plane which cuts it. The sliced portion is the required circle. So it may be a similar case with the ellipse. Something like one half of a cone being sliced by a plane.

 

[Gib Z]

It's probably not best to tell the OP to guess the formula from a pattern and say that its just "intuitive". One could just as well presume that since in one dimension x/a =1, and in 2 dimensions (x/a)^2 + (y/b)^2 = 1 that following the pattern into 3 dimensions is obviously (x/a)^3 + (y/b)^3 + (z/c)^3 = 1. We could probably think up a more stupid one as well.

Fatjjx - What you want to describe is an "Ellipsoid" - http://en.wikipedia.org/wiki/Ellipsoid

Basically, you start off with the equation of a Sphere, which is easy to derive using the Pythagorean theorem, and then you apply Linear Transformations, which in this case are just squeezing and stretching the sphere to make an ellipsoid.

 

[Defennder]

It's probably not best to tell the OP to guess the formula from a pattern and say that its just "intuitive". One could just as well presume that since in one dimension x/a =1, and in 2 dimensions (x/a)^2 + (y/b)^2 = 1 that following the pattern into 3 dimensions is obviously (x/a)^3 + (y/b)^3 + (z/c)^3 = 1. We could probably think up a more stupid one as well.

Fatjjx - What you want to describe is an "Ellipsoid" - http://en.wikipedia.org/wiki/Ellipsoid

Basically, you start off with the equation of a Sphere, which is easy to derive using the Pythagorean theorem, and then you apply Linear Transformations, which in this case are just squeezing and stretching the sphere to make an ellipsoid.

Did he mean ellipse or ellipsoid? The former is a 2D curve in 3D space whereas the latter is a 3D surface.

 

[Gib Z]

He already knew the equation of an ellipse, so I assumed he want to know the equation of an ellipsoid.

 

[Defennder]

I'm not too sure if he knows the formula of an ellipse curve in 3D space if the ellipse is not lying on the x-y plane. He may be referring to that. Unfortunate I don't know the general expression if it's not confined to the x-y plane either, but I found this:
http://mathforum.org/library/drmath/view/66054.html

 

[fatjjx]

indeed, what I want is just an ellipse in a 3D space, not the ellipsoid. I am doing the research of how the projection of a ellipse that is not parallel with the image plane will be. so I need the ellipse formula in 3D space. thank you for your response!

It's probably not best to tell the OP to guess the formula from a pattern and say that its just "intuitive". One could just as well presume that since in one dimension x/a =1, and in 2 dimensions (x/a)^2 + (y/b)^2 = 1 that following the pattern into 3 dimensions is obviously (x/a)^3 + (y/b)^3 + (z/c)^3 = 1. We could probably think up a more stupid one as well.

Fatjjx - What you want to describe is an "Ellipsoid" - http://en.wikipedia.org/wiki/Ellipsoid

Basically, you start off with the equation of a Sphere, which is easy to derive using the Pythagorean theorem, and then you apply Linear Transformations, which in this case are just squeezing and stretching the sphere to make an ellipsoid.

 

[HallsofIvy]

An ellipse is a curve- a one dimensional object. That means that an ellipse in 3 dimensions cannot be written as a single equation: each equation reduces the "degrees of freedom",i.e, dimension, by 1: 3- 1= 2 so any single equation in 3 dimensions gives a two dimensional object- as surface, such as the ellipsoid Gib Z gave. To define an ellipse in 3 dimensions you will need two equations in x, y, z. For example the equations $$x^2/a^2+ y^2\b^2= 1$$ and z= 0 define an ellipse in the xy-plane but $$x^2/a^2+ y^2/b^2= 1$$, z= 1 define an elliplse lying in the z= 1 plane. Ellipses at a tilt to any of the coordinate axes will be harder to write. You might try, for example, giving the equation of a cone such as [tex]z^2= x^2+ y^2[/itex] together with a plane like 3x+ 4y- z= 4. Together those define an ellipse. And, of course, you could write x, y, and z in terms of some single parameter t.

 

[symbolipoint]

According to HallsOfIvy discussion in post #10, my response in post #2 is wrong. My response would apply to the ellipsoid, not the ..., whatever you would call, ...? What do you call an ellipse projected perpendicularly to a plane?