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In particular, for three points in the plane (n = 2), the above matrix is square and the points are collinear if and only if its determinant is zero; since that 3 × 3 determinant is plus or minus twice the area of a triangle with those three points as vertices, this is equivalent to the statement that the three points are collinear if and only ...
They proved that the maximum number of points in the grid with no three points collinear is (). Similarly to Erdős's 2D construction, this can be accomplished by using points ( x , y , x 2 + y 2 {\displaystyle (x,y,x^{2}+y^{2}} mod p ) {\displaystyle p)} , where p {\displaystyle p} is a prime congruent to 3 mod 4 . [ 20 ]
The equations originate from the central projection of a point of the object through the optical centre of the camera to the image on the sensor plane. [1] The three points P, Q and R are projected on the plane S through the projection centre C x- and z-axis of the projection of P through the projection centre C
If the three points are collinear, R can be informally considered to be +∞, and it makes rigorous sense to define c(x, y, z) = 0. If any of the points x, y and z are coincident, again define c(x, y, z) = 0. Using the well-known formula relating the side lengths of a triangle to its area, it follows that
By extension, k points in a plane are collinear if and only if any (k–1) pairs of points have the same pairwise slopes. In Euclidean geometry, the Euclidean distance d(a,b) between two points a and b may be used to express the collinearity between three points by: [3] [4]
A complete quadrangle consists of four points, no three of which are collinear. In the Fano plane, the three points not on a complete quadrangle are the diagonal points of that quadrangle and are collinear. This contradicts the Fano axiom, often used as an axiom for the Euclidean plane, which states that the three diagonal points of a complete ...
Three or more collinear points, where the circumcircles are of infinite radii. Four or more points on a perfect circle, where the triangulation is ambiguous and all circumcenters are trivially identical. In this case the Voronoi diagram contains vertices of degree four or greater and its dual graph contains polygonal faces with four or more sides.
The basic condition for general position is that points do not fall on subvarieties of lower degree than necessary; in the plane two points should not be coincident, three points should not fall on a line, six points should not fall on a conic, ten points should not fall on a cubic, and likewise for higher degree. This is not sufficient, however.