Axonometry

of a semicircular archway. Axonometry is a graphical procedure belonging to descriptive geometry that generates a planar image of a three-dimensional object. The term "axonometry" means "to measure along axes", and indicates that the dimensions and scaling of the coordinate axes play a crucial role. The result of an axonometric procedure is a uniformly-scaled parallel projection of the object. In general, the resulting parallel projection is oblique (the rays are not perpendicular to the image plane); but in special cases the result is orthographic (the rays are perpendicular to the image plane), which in this context is called an orthogonal axonometry. In technical drawing and in architecture, axonometric perspective is a form of two-dimensional representation of three-dimensional objects whose goal is to preserve the impression of volume or relief. Sometimes also called rapid perspective or artificial perspective, it differs from conical perspective and does not represent what the eye actually sees: in particular parallel lines remain parallel and distant objects are not reduced in size. It can be considered a conical perspective conique whose center has been pushed out to infinity, i.e. very far from the object observed. The term axonometry is used both for the graphical procedure described below, as well as the image produced by this procedure.…

of a semicircular archway. Axonometry is a graphical procedure belonging to descriptive geometry that generates a planar image of a three-dimensional object. The term "axonometry" means "to measure along axes", and indicates that the dimensions and scaling of the coordinate axes play a crucial role. The result of an axonometric procedure is a uniformly-scaled parallel projection of the object. In general, the resulting parallel projection is oblique (the rays are not perpendicular to the image plane); but in special cases the result is orthographic (the rays are perpendicular to the image plane), which in this context is called an orthogonal axonometry. In technical drawing and in architecture, axonometric perspective is a form of two-dimensional representation of three-dimensional objects whose goal is to preserve the impression of volume or relief. Sometimes also called rapid perspective or artificial perspective, it differs from conical perspective and does not represent what the eye actually sees: in particular parallel lines remain parallel and distant objects are not reduced in size. It can be considered a conical perspective conique whose center has been pushed out to infinity, i.e. very far from the object observed. The term axonometry is used both for the graphical procedure described below, as well as the image produced by this procedure. Axonometry should not be confused with axonometric projection, which in English literature usually refers to orthogonal axonometry. The projection \overline P of a point P=(x,y,z) is determined in three sub-steps (the result is independent of the order of these sub-steps): starting at the point \overline O , move by the amount v_x\cdot x in the direction of \overline x , then move by the amount v_y\cdot y in the direction of \overline y , then move by the amount v_z\cdot z in the direction of \overline z and finally Mark the final position as point \overline P . In order to obtain undistorted results, select the projections of the axes and foreshortenings carefully (see below). In order to produce an orthographic projection, only the projections of the coordinate axes are freely selected; the foreshortenings are fixed (see :de:orthogonale Axonometrie). The parameters in the diagram at right (e.g. of the house drawn on graph paper) are: \alpha=135^\circ, \beta =90^\circ,\ v_y=v_z=1,\; v_x=1/\sqrt{2}\ . Hence it is a dimetric axonometry. The image plane is parallel to the y-z-plane and any planar figure parallel to the y-z-plane appears in its true shape. the foreshortenings are: v_x=0.5, \ v_y=v_z=1\ (dimetric axonometry) and the angles between the axes are: \alpha=132^\circ,\ \beta=97^\circ \ . These angles are marked on many German set squares. Advantages of an engineer projection: simple foreshortenings, a uniformly scaled orthographic projection with scaling factor 1.06, the contour of a sphere is a circle (in general, an ellipse) . For more details: see :de:Axonometrie. cabinet perspective: additionally choose \alpha=135^\circ (oblique) and v_x=0.5 (dimetric), cavalier perspective: additionally choose \alpha=135^\circ (oblique) and v_x=1 (isometric). military projection: additionally choose v_z=1 (isometric). Such axonometries are often used for city maps, in order to keep horizontal figures undistorted. For an isometric axonometry all foreshortenings are equal. The angles can be chosen arbitrarily, but a common choice is \alpha=\beta=\gamma=120^\circ . For the standard isometry or just isometry one chooses: v_x=v_y=v_z=1 (all axes undistorted) \alpha=\beta=\gamma=120^\circ\ . The advantage of a standard isometry: the coordinates can be taken unchanged, the image is a scaled orthographic projection with scale factor \sqrt{1.5}=1.225 . Hence the image has a good impression and the contour of a sphere is a circle. Some computer graphic systems (for example, xfig) provide a suitable raster (see diagram) as support. In order to prevent scaling, one can choose the unhandy foreshortenings v_x=v_y=v_z=\sqrt{2/3} (instead of 1) and the image is an (unscaled) orthographic projection. p = \begin{pmatrix}x\\y\\z\end{pmatrix} \mapsto f(p) = \begin{pmatrix}x'\\y'\end{pmatrix} which maps the coordinate vector p \in \mathbb R^3 of a general point P in space to the coordinate vector in \mathbb R^2 of the image point \overline P. The map f is linear and can therefore be represented by its 2\!\times\!3 transformation matrix A\colon f(p) = A\!\ p (matrix times vector). The three columns of A are the image vectors f(e_x),f(e_y),f(e_z) \in \mathbb R^2 of the standard basis vectors e_x = \begin{pmatrix}1\\0\\0\end{pmatrix},\quad e_y = \begin{pmatrix}0\\1\\0\end{pmatrix},\quad e_z = \begin{pmatrix}0\\0\\1\end{pmatrix} and thus describe the \overline x, \overline y und \overline z image axes. They do not lie on a common straight line through the origin, i.e., the real 2\!\times\!3 matrix A has rank 2. Conversely, each such matrix generates an axonometry. A = \begin{pmatrix}-v_x\sin\alpha& v_y\sin\beta& 0\\ v_x\cos\alpha& v_y\cos\beta& v_z\end{pmatrix}. It is assumed that the y'\text{-} axis and the \overline z image axis point upwards and the x'\text{-} axis points to the right. 2. In particular, v_x = v_y = v_z = 1 and \alpha = \beta = 120^\circ give the standard isometry with the matrix A = \begin{pmatrix}-\frac{\sqrt3}2& \frac{\sqrt 3}{2}& 0\\-\frac12& -\frac12 &1\end{pmatrix}. For instance, the vertex p = (1,1,1)^\mathsf{T} of the unit cube is mapped to f(p) = A\!\ p = (0,0)^\mathsf{T}, as is the opposite vertex (0,0,0)^\mathsf{T}, where {}^\mathsf{T} denotes transposition. As mentioned above, standard isometry is an orthographic projection scaled by the factor \textstyle\sqrt{3/2}\,. 3. Multiplying the last matrix by the reciprocal \textstyle\sqrt{2/3} thus yields the transformation matrix A = \begin{pmatrix}-\frac1{\sqrt2}& \frac1{\sqrt2}& 0\\ -\frac1{\sqrt6}& -\frac1{\sqrt6}& \sqrt{\frac23}\end{pmatrix} of an (unscaled) orthographic projection. Incidentally, in its transpose A^\mathsf{T}, both columns e_x' = \begin{pmatrix}-\frac1{\sqrt2}\\ \frac1{\sqrt2}\\[0.3ex] 0\end{pmatrix},\quad e_y' = \begin{pmatrix}-\frac1{\sqrt6}\\ -\frac1{\sqrt6}\\ \sqrt{\frac23}\end{pmatrix} are orthonormal, i.e., e_x'\cdot e_y' = 0 and e_x'\cdot e_x' = e_y'\cdot e_y' = 1. This characterizes orthographic projections, as is shown exemplarily hereinafter. 4. Another orthographic projection is constructed, now conversely, starting with the orthonormal vectors e_x' = \frac17 \begin{pmatrix}6\\-3\\-2\end{pmatrix},\quad e_y' = \frac17 \begin{pmatrix}3\\2\\6\end{pmatrix}. These two together with e_z' = e_x'\times e_y' and O define another Cartesian coordinate system in space. A general point P in space now has another coordinate vector p' = (x',y',z')^\mathsf{T} besides p = (x,y,z)^\mathsf{T}. For these, always E\!\ p' = p with the 3\!\times\!3 change-of-basis matrix E = \big(e_x'\,\big|\,e_y'\,\big|\,e_z'\big). As E is orthogonal, E^{-1} = E^\mathsf{T} and consequently p' = E^\mathsf{T}p. The orthographic projection, which projects perpendicularly onto the x'\text{-}y'\text{-} plane as the drawing plane, corresponds to omitting the bottom z'\text{-} coordinate of p'. Its transformation matrix is obtained accordingly from E^\mathsf{T} by omitting the bottom row: A = \frac17\, \begin{pmatrix}6&-3&-2\\ 3&2&6\end{pmatrix}. This can be used to calculate points in the adjacent drawing of the unit cube. For instance, the vertex p = (1,1,1)^\mathsf{T} is mapped to f(p) = A\!\ p = \big(\tfrac17,\tfrac{11}7\big){}^\mathsf{T}.

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