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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 ...
Let x, y, and z refer to a coordinate system with the x- and y-axis in the sensor plane. Denote the coordinates of the point P on the object by ,,, the coordinates of the image point of P on the sensor plane by x and y and the coordinates of the projection (optical) centre by ,,.
In physics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear vectors, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding vectors. According to the theorem,
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
In three-dimensional space, two linearly independent vectors with the same initial point determine a plane through that point. Their cross product is a normal vector to that plane, and any vector orthogonal to this cross product through the initial point will lie in the plane. [ 1 ]
Illustration showing how to find the angle between vectors using the dot product Calculating bond angles of a symmetrical tetrahedral molecular geometry using a dot product. In Euclidean space, a Euclidean vector is a geometric object that possesses both a magnitude and a direction. A vector can be pictured as an arrow.
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 ...
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.