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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 ...
In projective geometry, the harmonic conjugate point of a point on the real projective line with respect to two other points is defined by the following construction: Given three collinear points A, B, C , let L be a point not lying on their join and let any line through C meet LA, LB at M, N respectively.
Simply, a collineation is a one-to-one map from one projective space to another, or from a projective space to itself, such that the images of collinear points are themselves collinear. One may formalize this using various ways of presenting a projective space. Also, the case of the projective line is special, and hence generally treated ...
However, in a pappian projective plane a conic is a circle only if it passes through two specific points on the line at infinity, so a circle is determined by five non-collinear points, three in the affine plane and these two special points. Similar considerations explain the smaller than expected number of points needed to define pencils of ...
the points AB ∩ ab, AC ∩ ac and BC ∩ bc are collinear. The points A, B, a and b are coplanar (lie in the same plane) because of the assumed concurrency of Aa and Bb. Therefore, the lines AB and ab belong to the same plane and must intersect. Further, if the two triangles lie on different planes, then the point AB ∩ ab belongs to
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
Points that are incident with the same line are said to be collinear. The set of all points incident with the same line is called a range. If P 1 = (x 1, y 1, z 1), P 2 = (x 2, y 2, z 2), and P 3 = (x 3, y 3, z 3), then these points are collinear if and only if
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.