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I've got a normal vector of a plane and an initial 3D coordinate. Another point exists in another location not on the plane. I want to find a way to calculate the set of curved paths starting initially at the coordinate on the plan going in the direction of the normal vector to the other coordinate off the plane.

I'm looking for a more general solution to the equation that would have one solution be a quadrant of circle with some radius r.

I think one important point I was missing initially in my question is the start can be curved and the finish at the "other" point is in the direction parallel to the plane.

I'm coming from a pretty naive position on this question so please let me know if it is too ambiguous to ask in the first place.

EDIT: Right now I am referencing Smooth curve from two known points with known normal vectors. I am trying to solve the "equation" which creates the curved arc between 2 points. Where the curve starts in the direction of n1 at point p1.

EDIT2:

Image of the inner point

Image of the spheres that make the plane

From the images above you can see in the first one I've got an inner set of spheres in yellow (atoms) that I average to give me a 3 dimensional coordinate. Also in both pictures there are blue spheres (atoms) I cluster them into 4 separate groups and get the average of each group. I then have 4 points corresponding to the 4 clustered blue sphere and 1 point for the clustered yellow spheres.

I calculate the best fit plane for the 4 blue sphere clusters. I also calculate the middle between all of the blue sphere. Using the middle of the blue spheres as the starting point, I want a curve initially starting in the direction of the normal of the plane then continuing to curve until it reaches the point in the yellow cluster of spheres. Preferably a smooth curve with the final direction of the curve pointing in the same direction as the plane. Hope this makes sense.

I will work on generating the plane with the points in question to simplify if needed.

EDIT3: Here is an image of the final geometry without the ending point. The starting point is in red. I need the curve to go in the direction of the green where red to green is the normal vector of the plane.

Image of the plane without finishing dot

In this picture the red dot off a ways from the plane is the end point. Ignore the green dot it gets skewed for whatever reason in this graph.

image of the plane with finish dot

Le Bonez
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  • What do you mean by " and a coordinate on that plane" ? A coordinate system $(O, \vec{i}, \vec{j})$ on that plane ? – Jean Marie Mar 13 '21 at 19:04
  • I might've answered my own question by noticing a concept called Bézier curve. Initial reading makes me think it might not be general enough but is a good start. – Le Bonez Mar 13 '21 at 19:04
  • Bezier curves are a wonderful tool... – Jean Marie Mar 13 '21 at 19:05
  • @JeanMarie I see what you mean. It is a point which defines where the plane is located given some origin not on the plane. Is that what you were asking? – Le Bonez Mar 13 '21 at 19:08
  • All right, I understand: the plane is defined by one of its points and a normal vector to it. – Jean Marie Mar 13 '21 at 19:37
  • Can you show us a picture with the plane and the point and some of the curves in the family of curves you are interested in? Do you really want a family of curves, or just one with some particular properties? – Ethan Bolker Mar 13 '21 at 20:15
  • @EthanBolker I have added the images from the problem itself which is a cluster of atoms. I'm trying to figure out the best way to generate a plane and the 6 points that are important. – Le Bonez Mar 13 '21 at 20:56
  • I think we need just the final geometry: a three dimensional picture of the points and the plane, not the spheres you use to generate them. – Ethan Bolker Mar 13 '21 at 20:59
  • @EthanBolker I have added two images showing the plane and the dots associated – Le Bonez Mar 13 '21 at 21:12

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