# Eelipsoid and line intersection

• ppmko
In summary, the problem involves finding the coordinates of a point P(x,y,z) on an ellipsoid with center (0,0,0) that is intersected by a line connecting the center to a point A(1,2,3) inside the ellipsoid. The equation of the ellipsoid is expressed as x^2/a^2+y^2/b^2+z^2/c^2=1 and the line is described in parametric form as x=t, y=2t, z=3t. By substituting these values and solving for t, the coordinates of P(x,y,z) can be found. However, the distance from the center to P should equal the sum of the distances from the center
ppmko
Homework Statement

I have an ellipsoid with center (000). There is a point A inside the ellipsoid with known coordinates(1,2,3) I draw a line from center to point A and extend it to cut the ellipsoid on on point p(x,y,z).

## Homework Equations

I want to find the coordinates of point P(x,y,z)

## The Attempt at a Solution

The equation of ellipsoid for p is x^2/a^2+y^2/b^2+z^2/c^2=1
i have the values of a,b and c
i want to know if the ellipsoid equation is applicable to coordinates of A and coordinates of P

and how can i create equation using the coordinates of p with the coordinates of A
by equation of line method as both the points lie on a straight line with one end on (000) as the third point.

Express the line through (0,0,0) and (1,2,3) in parametric form, i.e. x=t, y=2t, z=3t. Put that into the ellipsoid equation. Solve for t.

Thank you very much . I tried that method but the problem I am facing is say my point A (1,2,3) is inside the ellipsoid and center(000) and the point on the ellipsoid i solve using
x^2/a2+y^2/b2+Z^2/c^2=1 by substituting x=t,y=2t,z=3t and solving for t. But the distance between center and point (x,y,z) should be equal to the sum of the dist between center and A and A and point(x,y,z) . that is not matching .i am using the formula for dist between two pints say(x1,y1,z1) and (x2,y2,z2) as sqrt(x2-x1)^2+(y2-y1)^2+(z2-z1)^2.
can you tell me where i am going wrong

ppmko said:
Thank you very much . I tried that method but the problem I am facing is say my point A (1,2,3) is inside the ellipsoid and center(000) and the point on the ellipsoid i solve using
x^2/a2+y^2/b2+Z^2/c^2=1 by substituting x=t,y=2t,z=3t and solving for t. But the distance between center and point (x,y,z) should be equal to the sum of the dist between center and A and A and point(x,y,z) . that is not matching .i am using the formula for dist between two pints say(x1,y1,z1) and (x2,y2,z2) as sqrt(x2-x1)^2+(y2-y1)^2+(z2-z1)^2.
can you tell me where i am going wrong

What did you use for a, b and c, what did you get for t and hence for (x,y,z)? Of course you should find that the distance from O to A plus the distance from A to (x,y,z) should equal the distance from O to (x,y,z). But it's impossible to say what you are doing wrong until you tell us what you did.

I used a=1
b=2
c=3
for coordinates of A(1,2,3) and center(0,0,0)

now my equaation becomes for point p(x,y,z) on ellipsoid
x=0+(1-0)t=t
y=0+(2-0)t=2t
z=0+(3-0)t=3t

put in ellipsoid equation
t^2/1+4t^2/4+9t^2/9=1

3t^2=1
t=sqrt(1/3)=.57

x=.57
y=1.15
z=1.71

now pa=sqrt[(x-1)^2+(y-2)^2+(z-3)^2]=1.603
point A and center(000)=3.74
whereas my p to center is 3.30
i cannoot understand where i am going wrong

I don't think your distance from p to the center is right. But more importantly, if you choose a=1, b=2, and c=3 then your selected point A=(1,2,3) is OUTSIDE of the ellipsoid. I thought you were going to put it inside.

i need the point A to be inside . I guess I should have given the arbitraty values for the axis as a=3 b=2 and c=1 . I hope a,b,c are the semimajor axis,semiminor axis and the z axis repectively in the ellipsoid equation . and the coordinates of A(1,2,3) are separate.
the point on the ellipsoid will have x^2/a^2+y^2/b^2+z^2/3^3=1.where the x=t,y=2t,z=3t as the line touching the point on ellipsoid pwill pass through the center(000) and A(1,2,3) located inside the ellipsoid.
I tried that too but still it is not matching.

Sure, a, b and c are the axes of the ellipse. But if you want A(1,2,3) to be inside the ellipse, you need to make a, b and c larger than 1, 2 and 3.

## 1. What is an ellipsoid?

An ellipsoid is a three-dimensional geometric shape that resembles a flattened sphere. It is defined by three radii, known as the semi-major axis, semi-minor axis, and semi-intermediate axis.

## 2. How is an ellipsoid represented mathematically?

An ellipsoid is typically represented by an equation that relates the coordinates of a point on the surface of the shape to the three radii. This equation is known as the ellipsoid's implicit equation.

## 3. What is the equation for the intersection of an ellipsoid and a line?

The equation for the intersection of an ellipsoid and a line is a quadratic equation that can be solved to find the points where the line intersects the surface of the ellipsoid. This equation involves the coordinates of the line's direction vector and the coefficients of the ellipsoid's implicit equation.

## 4. How can the intersection of an ellipsoid and a line be used in scientific research?

The intersection of an ellipsoid and a line can be used in various fields such as geodesy, remote sensing, and geophysics. It is particularly useful in determining the position and orientation of objects in three-dimensional space, such as satellites or geological structures.

## 5. Are there any special cases in which the intersection of an ellipsoid and a line may not exist?

Yes, there are two special cases in which the intersection of an ellipsoid and a line may not exist. The first case is when the line is parallel to the surface of the ellipsoid, and the second case is when the line is tangent to the surface of the ellipsoid at a single point.

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