In 3d space, is there any way to project a vector onto a plane, but along the UP direction (0,1,0) instead of the plane normal? If so, how do I do that and what is it called?

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canhaulatlt · Accepted Answer

If I understand you correctly, you want to take an arbitrary vector             v      →       and decompose it as             v      →        =            w      →        +  α            u      →       where             w      →        lies in some plane with normal             n      →        (i.e.,             w      →        ⋅            n      →        =  0), and             u      →       is a known vector, not necessarily equal to             n      →      Since by assumption we have             v      →        =            w      →        +  α            u      →       and             w      →        ⋅            n      →        =  0, taking the dot product of both sides with             n      →       yields            v      →        ⋅            n      →        =  α            u      →        ⋅            n      →          ⇒    α  =                              v          →                    ⋅                        n          →                                              u          →                    ⋅                        n          →                      .This then implies that the desired vector             w      →       will be                                                        w              →                                =                                    v              →                                −                      (                                                                                v                    →                                                  ⋅                                                      n                    →                                                                                                                    u                    →                                                  ⋅                                                      n                    →                                                                        )                                              u              →                                .                    Note that in the case where             u      →        =            n      →      , this reduces to the standard orthogonal projection equation. Note also that in the case where             u      →        ⋅            n      →        =  0, this equation doesn&#039;t make any sense; this corresponds to the case where you&#039;re projecting along an axis lying parallel to the chosen plane, and so the desired vector             w      →        will not exist unless             v      →       already lies in that plane.

omgespit9q · Accepted Answer

A picture would help, but this is what I think you are asking. Given the plane   a  x  +  b  y  +  c  z  =  d and the point   (  p  ,  q  ,  r  ), you want to move the point in the direction   (  0  ,  1  ,  0  )  until it meets the plane.Well that move will change only the value of  q, To find the new value, solve  a  p  +  b  ?  +  c  r  =  dfor the value of ?There will be no solution if (0,1,0) is parallel to the plane (unless the point is in the plane to begin with).In more generality, you are asking for the intersection of a line (through a given point in a given direction) with a plane.

In 3d space, is there any way to project a vector onto a plane, but along the UP direction (0,1,0) instead of the plane normal? If so, how do I do that and what is it called?

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