Convert equation of a line on a plane to R3 equation

[swraman]

Hi,

I have the equation of a plane P in R3.

I have an equation f(x) of a line that exists on that plane, using an arbitrary origin/axis on the plane.

I know the corresponding R3 coordinates of a point of f(x).

Intuition tells me that I have all the info I need to calculate the equation to the line in R3, but I don't know exactly how.

Any help appreciated.

Thanks

 

[mathman]

Is the origin/axis for f(x) known in terms of R3 coordinates? If so then you don't need the R3 coordinates of a point of f(x).

Alternatively if the origin/axis is not specified, then you need two points on the line. The plane equation is not needed.

 

[swraman]

Is the origin/axis for f(x) known in terms of R3 coordinates? If so then you don't need the R3 coordinates of a point of f(x).

Alternatively if the origin/axis is not specified, then you need two points on the line. The plane equation is not needed.

I have 3 points (in R3 coordinates) that the are on the function. It is aparabola, so I assume you need 3 points to construct.

so taking into consideration what you said, i guess my question is now:

how do you equation to a parabola in the form:

x(t)
y(t)
z(t)

from 3 points in R3?

Thanks

 

[mathman]

In general you would need four points to determine a parabola. The general quadratic curve has six parameters, but these are unique only up to a constant multiplier, leaving five free parameters. Making it a parabola imposes one condition, reducing the number of free parameters to four.