- #1

- 545

- 77

_{i}in a space of dimension N, aside from degenerate intersections (e.g., a pair of spheres that touch at a single point), the dimension of the net intersection locus L is:

L = N - ∑ ( N - D

_{i}) = ( ∑ D

_{i}) - N ( M - 1 )

pair of lines in a plane: L = ( 1 + 1 ) - { 2 } ( { 2 } - 1 ) = 0 -> point

pair of planes in 3-D space: L = ( 2 + 2 ) - { 3 } ( { 2 } - 1 ) = 1 -> line

3 planes in 3-D space: L = ( 2 + 2 + 2 ) - { 3 } ( { 3 } - 1 ) = 0 -> point

plane & line in 3-D space: L = ( 2 + 1 ) - { 3 } ( { 2 } - 1 ) = 0 -> point

And for the case of linear loci, this can be proven in linear algebra. Then for the case of generally shaped loci, there is a topological correspondence between that loci and a linear loci of the same dimension (i.e., the linear loci can be stretched into whatever shape the general loci is).

Is this accurate?