China 
Africa 
Explore chapters and articles related to this topic
Geometry
Published in Leslie Hogben, Richard Brualdi, Anne Greenbaum, Roy Mathias, Handbook of Linear Algebra, 2006
In the real projective plane, consider the six points A= [1:2:1], B = [2:0:1],C = [5:5:1], A′= [2:4:1], B′= [4:0:1], and C′= [10:10:1]. Identifying the projective point with homogenous coordinates [a:b:1] with the ordinary Euclidean point [ab], as in Example 6 above, it can be seen by a simple diagram that the triangles ABC and A′B′C′ are perspective from the origin, since all lines AA′, BB′, and CC′ pass through this point. Thus, in the real projective plane these triangles are perspective from the point [0:0:1]. Desargues’ Theorem, therefore, asserts that the points of intersection of the lines AB and A′B′, AC and A′C′, BC and B′C′ will lie on a line. A simple calculation shows that these three points, in homogenous coordinates, have third component zero, so these points are indeed collinear. This corresponds to the fact that as Euclidean lines there are no points of intersection — the corresponding sides of the triangles are parallel in pairs. The points with third component zero are the “ideal points” that have been added to Euclidean geometry to form the real projective plane.
Systems of Linear Equations
Published in James R. Kirkwood, Bessie H. Kirkwood, Linear Algebra, 2020
James R. Kirkwood, Bessie H. Kirkwood
In the next example we find the equation of a plane that passes through three given points. There will be such a plane if and only if the points are not collinear. If there is a plane, it will have an equation of the form ax+by+cz=d.
Fabrication of parts with heterogeneous structure using material extrusion additive manufacturing
Published in Virtual and Physical Prototyping, 2021
Aggelos Vassilakos, John Giannatsis, Vassilis Dedoussis
The final step in the methodology proposed in this work involves the creation of the layer’s infill and border (shell) deposition paths. The layer infill path is first determined by joining individual cell paths following the cell visiting sequence. The aggregate/merged infill path contains several redundant points (tile centres) that are mostly found at the joining areas between cells. These redundant points are identified via a simple routine that checks whether three consecutive points are collinear. This operation does not affect the shape of the infill path and reduces the size of the G-code file. This is beneficial in terms of both computer processing effort and fabrication time; relatively longer path segments are implied, allowing thereby for higher deposition speeds (feed rates).