- #1
fatjjx
- 3
- 0
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!
thank you!
symbolipoint said: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?
Did he mean ellipse or ellipsoid? The former is a 2D curve in 3D space whereas the latter is a 3D surface.Gib Z said: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.
Gib Z said: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.
The general formula for an ellipse in 3D space is (x/a)^2 + (y/b)^2 + (z/c)^2 = 1, where 'a', 'b', and 'c' are constants representing the lengths of the semi-major axis, semi-minor axis, and the axis perpendicular to the plane of the ellipse, respectively.
The equation for an ellipse in 3D space is similar to that in 2D space, except it includes an additional term for the z-axis. In 2D space, the equation is (x/a)^2 + (y/b)^2 = 1, where 'a' and 'b' are the lengths of the semi-major and semi-minor axes, respectively.
The constants 'a', 'b', and 'c' represent the lengths of the semi-major axis, semi-minor axis, and the axis perpendicular to the plane of the ellipse, respectively. These values determine the size and orientation of the ellipse in 3D space.
To graph an ellipse in 3D space using its formula, you first need to identify the values of 'a', 'b', and 'c' and determine the center point of the ellipse. Then, plot points on the x, y, and z axes using the values of 'a', 'b', and 'c' as the radius and the center point as the origin. Finally, connect the points to form the ellipse.
Yes, the formula for an ellipse in 3D space can be used for all ellipses, regardless of their size, orientation, or position in space. However, the values of 'a', 'b', and 'c' may vary for different ellipses, depending on their specific characteristics.