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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 ...
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 ...
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. Equivalently for three points in a plane, the points are collinear if and only if the slope between one pair of points equals the slope between any other pair of points (in which case the slope ...
The interval AB is the segment AB and its end points A and B. The ray A/B (read as "the ray from A away from B") is the set of points P such that [PAB]. The line AB is the interval AB and the two rays A/B and B/A. Points on the line AB are said to be collinear. An angle consists of a point O (the vertex) and two non-collinear rays out from O ...
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
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 ...
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
Similarly, three generic points in the plane are not collinear; if three points are collinear (even stronger, if two coincide), this is a degenerate case. This notion is important in mathematics and its applications, because degenerate cases may require an exceptional treatment; for example, when stating general theorems or giving precise ...