Search results
Results From The WOW.Com Content Network
This happens if and only if the triangle vertices aren't collinear and the ray isn't parallel to the plane. The algorithm can use Cramer's Rule to find the t {\displaystyle t} , u {\displaystyle u} , and v {\displaystyle v} values for an intersection, and if it lies within the triangle, the exact coordinates of the intersection can be found by ...
In any geometry, the set of points on a line are said to be collinear. In Euclidean geometry this relation is intuitively visualized by points lying in a row on a "straight line". However, in most geometries (including Euclidean) a line is typically a primitive (undefined) object type , so such visualizations will not necessarily be appropriate.
Let X be an affine space over a field k, and V be its associated vector space. An affine transformation is a bijection f from X onto itself that is an affine map; this means that a linear map g from V to V is well defined by the equation () = (); here, as usual, the subtraction of two points denotes the free vector from the second point to the first one, and "well-defined" means that ...
Barycentric coordinates are strongly related to Cartesian coordinates and, more generally, affine coordinates.For a space of dimension n, these coordinate systems are defined relative to a point O, the origin, whose coordinates are zero, and n points , …,, whose coordinates are zero except that of index i that equals one.
In this algorithm, one recursively draws a line to split the vertices into two sets. The Delaunay triangulation is computed for each set, and then the two sets are merged along the splitting line. Using some clever tricks, the merge operation can be done in time O( n ) , so the total running time is O( n log n ) .
[1]: p. 23 From this, every straight line has a linear equation homogeneous in x, y, z. Every equation of the form l x + m y + n z = 0 {\displaystyle lx+my+nz=0} in real coefficients is a real straight line of finite points unless l : m : n is proportional to a : b : c , the side lengths, in which case we have the locus of points at infinity.
The projective linear group of n-space = (+) has (n + 1) 2 − 1 dimensions (because it is (,) = ((+,)), projectivization removing one dimension), but in other dimensions the projective linear group is only 2-transitive – because three collinear points must be mapped to three collinear points (which is not a restriction in the projective line ...
A permutation of the seven points that carries collinear points (points on the same line) to collinear points is called a collineation or symmetry of the plane. The collineations of a geometry form a group under composition, and for the Fano plane this group (PΓL(3, 2) = PGL(3, 2)) has 168 elements.