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enter image description here

Consider the picture which I have provided in the link. In the picture I have connected all the points to form the plane that is going through the cube. However this isn't all of the plane cutting the cube. There is some missing. How would one construct the rest of the plane when you're only given the points in the picture? In other words, we want to find the cross section of the plane through the cube. What makes this difficult is the fact that none of the points lie on the same plane. If they did, we could extend a line through them and construct the whole cross section. However, in this picture all the points are on skewed lines.

I have no idea how the construction would go. Anyone have ideas?

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  • $\begingroup$ Oh, ImageShack. Brings back memories... $\endgroup$
    – Shahar
    Commented Sep 3, 2014 at 23:23
  • $\begingroup$ To clarify your question: If you have three points, you can construct the plane in which all of the lie. However, this plane is of infinite extent i.e. it exceeds the cube. What it sounds like you want is just the cross-section, i.e. the region of this plane which falls in the cube. Is that correct? $\endgroup$
    – Semiclassical
    Commented Sep 4, 2014 at 0:50
  • $\begingroup$ Exactly. Just the region that falls in the cube. $\endgroup$
    – Nick Freeman
    Commented Sep 4, 2014 at 1:25
  • $\begingroup$ Your problem is essentially that of clipping a plane against a cube. The boundary of the region is a set of line segments in space. Are you interested in an analytical method? $\endgroup$
    – user_of_math
    Commented Sep 4, 2014 at 4:50
  • $\begingroup$ Is there no one here that can do this with pure geometrical construction (pure geometric drawing) and not using analytical method ? $\endgroup$
    – Tobi123
    Commented Aug 10, 2019 at 22:03

1 Answer 1

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… none of the points lie on the same plane.

This is confusing. Three points define a plane. So you have a uniquely defined plane in space, and that plane intersects the cube. You are looking for the intersection of that plane with the cube. To obtain that, I'd intersect the plane with each edge of the cube in turn, and if the intersection is inside the edge (as opposed to its extension beyond the vertices of the cube), then you have a corner of the intersection figure. The complete intersection figure is then the convex hull of all these corners.

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  • $\begingroup$ Your idea sounds good but how would you construct it? You know what I mean. Or maybe you are explaining how to construct it, but I am not following, which, if that is the case, I apologize. $\endgroup$
    – Nick Freeman
    Commented Sep 4, 2014 at 16:13
  • $\begingroup$ @NickFreeman: What exactly do you mean by “construction”? On the one hand, the classical ruler-and-compass construction interpretation of the term doesn't apply to 3D, except if you have to do this in the projection on your paper. On the other hand, if you have coordinates, “computation” might be the more appropriate term. So what setup are you dealing with here? $\endgroup$
    – MvG
    Commented Sep 4, 2014 at 16:19
  • $\begingroup$ @MvG He meant something like this : 3.bp.blogspot.com/-uure6odxwqE/Wlg9bn1gPeI/AAAAAAAAHjA/… $\endgroup$
    – Tobi123
    Commented Aug 10, 2019 at 21:15
  • $\begingroup$ Given 3 points (no coordinate known) how do you find another points of intersection of that plane defined by those 3 given points with the cube so that he can construct the cross section, and this must be done without calculation (pure geometrical construction), this subject has been bugging me too because there's no tutorial for it in google at all (at least not in english) $\endgroup$
    – Tobi123
    Commented Aug 10, 2019 at 21:18
  • $\begingroup$ How do you do a pure geometrical construction in 3d? Any single planar representation will lack full information about the 3d location of points. Do you have your 3d situation projected onto multiple planes at the same time? Then this should be doable. $\endgroup$
    – MvG
    Commented Aug 11, 2019 at 9:00

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