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If the normal of the viewing plane (the camera direction) is parallel to one of the primary axes (which is the x, y, or z axis), the mathematical transformation is as follows; To project the 3D point , , onto the 2D point , using an orthographic projection parallel to the y axis (where positive y represents forward direction - profile view ...
With an axonometric projection, the scale of an object does not depend on its location (i.e., an object in the "foreground" has the same scale as an object in the "background"); consequently, such pictures look distorted, as human vision and photography use perspective projection, in which the perceived scale of an object depends on its ...
The black dimensions are the true lengths as found in an orthographic projection. The red dimensions are used when drawing with the isometric drawing method. The same 3D shapes drawn in isometric projection would appear smaller; an isometric projection will show the object's sides foreshortened, by approximately 80%.
The point ¯ is the projection of a point = (,,) onto the projection plane Π. The foreshortenings are v x {\displaystyle v_{x}} , v y {\displaystyle v_{y}} and v z {\displaystyle v_{z}} . Pohlke's theorem is the basis for the following procedure to construct a scaled parallel projection of a three-dimensional object: [ 1 ] [ 2 ]
Similarly in 3 dimensions, the duality relation holds between points and planes, allowing any theorem to be transformed by swapping point and plane, is contained by and contains. More generally, for projective spaces of dimension N, there is a duality between the subspaces of dimension R and dimension N − R − 1.
Two figures in a plane are perspective from a point O, called the center of perspectivity, if the lines joining corresponding points of the figures all meet at O. Dually , the figures are said to be perspective from a line if the points of intersection of corresponding lines all lie on one line.
The plane z = 0 runs through the center of the sphere; the "equator" is the intersection of the sphere with this plane. For any point P on M, there is a unique line through N and P, and this line intersects the plane z = 0 in exactly one point P ′, known as the stereographic projection of P onto the plane.
Comparison of several types of graphical projection, including elevation and plan views. To render each such picture, a ray of sight (also called a projection line, projection ray or line of sight) towards the object is chosen, which determines on the object various points of interest (for instance, the points that are visible when looking at the object along the ray of sight); those points of ...