Equations for tangent & normal at P2 of circle P1 P2 P3?

[CosmicVoyager]

Greetings,

Given three points P1 P2 P3 on a circle in x,y,z coordinates, I am trying to figure out how to get the tangent and normal at P2.

Anyone?

Thanks

 

[tiny-tim]

Hi CosmicVoyager!

Well, that means you first need to find the centre of the circle …

what lines do you think that will be on? ​

 

[CosmicVoyager]

Hi CosmicVoyager!

Well, that means you first need to find the centre of the circle …

what lines do you think that will be on? ​

The center of the circle won't be on any of the lines between the points. It is opposite the their normals?

 

[HallsofIvy]

If you construct the perpendicular bisectors of the lines between the points, they will intersect at the center of the circle.

Once you know that, construct the line from that center to each point. That line itself will be normal to the circle at the point. Constructing the line perpendicular to that line at the point gives you the tangent to the circle at that point.

 

[CellCoree]

for reaching out. The equations for the tangent and normal at P2 of a circle can be found using the concept of derivatives. The tangent at a point P2 is a line that touches the circle at that point, and its slope is equal to the derivative of the circle's equation at P2. The normal at P2 is a line that is perpendicular to the tangent at P2, and its slope is equal to the negative reciprocal of the tangent's slope.

To find the equations for the tangent and normal at P2, you can start by finding the equation of the circle using the coordinates of P1, P2, and P3. This can be done by using the distance formula to find the radius of the circle and then plugging it into the general equation of a circle (x-h)^2 + (y-k)^2 = r^2, where (h,k) is the center of the circle.

Next, you can find the derivative of the circle's equation with respect to x, which will give you the slope of the tangent at P2. This can be done using the power rule for derivatives.

Finally, you can find the slope of the normal at P2 by taking the negative reciprocal of the tangent's slope. Once you have the slopes, you can find the equations of the tangent and normal lines using the point-slope form.

I hope this helps. Let me know if you have any further questions. Happy experimenting!