Use the following functions defined in the GraphicsFunctions.mthZG4ZTI utility file to project 3D rectangular coordinate vectors into two dimensions so parametrically defined space curves and surfaces can be plotted in 2D-plot windows.  Note that 3D rectangular coordinate vectors can be plotted directly in 3D-plot windows (for details, see the Insert > Plot command72HLS1).  The Coordinate System Conversion FunctionsCoordinate_System_Conversion_Functions can be used to convert surfaces parametrically defined in other coordinate systems to rectangular coordinate vectors.


Images of three dimensional objects are projected using an isometric (trimetric) projection.  That is images are projected perpendicular to the screen as if viewed from a great distance through a telephoto lens.  For example, the isometric projection of the front surface of a cube does not appear larger (or smaller) than the back surface.


The 2D and 3D origins coincide.  The positive 3D x-axis points down to the left and out of the screen.  The positive 3D y-axis points symmetrically down to the right and out of the screen.  The positive 3D z-axis points symmetrically up and out of the screen.  This orientation is close to the default orientation in a 3D-plot window.  However, hidden lines on parametric surfaces are not removed, so the effect is a wire frame rather than a scored opaque surface.


axes  is assigned to a 3 x 2 matrix that parametrically specifies the isometric projections of the x, y, and z axes of the 3D rectangular (Cartesian) coordinate system.  When axes is plotted in a 2D-plot window, you are successively prompted for the minimum and maximum values of the x-axis, the y-axis, and then the z-axis.  These axes are drawn in that order from these minimum to maximum values.  For example, use 0 as a minimum value if you only want to see the positive axis.  Press Ctrl+Enter or Ctrl+J instead of Enter to make the same minimum and maximum values apply to the remaining axes.


A 3D coordinate vector can represent a point, a space curve, or a parametric surface.  The three elements of the vector respectively give the x, y, and z coordinates of the point, curve, or surface.  If the coordinates are constants, the coordinate vector represents a point.  If the coordinates depend on one parameter, the vector represents a space curve.  If the coordinates depend on two parameters, the vector represents a parametric surface.


ISOMETRIC(v) simplifies to the 2D isometric projection of the 3D coordinate vector v representing a space curve.


ROTATE_X(α) simplifies to a matrix A such that A·v rotates the 3D coordinate vector v through an angle of α radians about the x-axis, counterclockwise when viewed from the positive x-axis toward the origin.


ROTATE_Y(α) simplifies to a matrix A such that A·v rotates the 3D coordinate vector v through an angle of α radians about the y-axis, counterclockwise when viewed from the positive y-axis toward the origin.


ROTATE_Z(α) simplifies to a matrix A such that A·v rotates the 3D coordinate vector v through an angle of α radians about the z-axis, counterclockwise when viewed from the positive z-axis toward the origin.


ISOMETRICS(v, s, s0, sm, m, t, t0, tn, n) approximates to a vector of n+1 two-column matrices that plots in a 2D-plot window as the isometric projection of the parametric surface specified by 3D coordinate vector v which is defined in terms of parameters s and t.  This vector is evaluated on a grid of points with s varying from s0 through sm in m equal increments and t varying from t0 through tn in n equal increments.


To produce a plot of line segments along lines of constant t, use the 2D-plot window Set > Coordinate System command162L784 to select the rectangular coordinate system and the Options > Display > Points command7CKH0ZT to select connected points that are medium or large in size.


To make the dots and lines in such plots all the same color, turn off the option to automatically change plot colors (see the Options > Display > Color commandJDDXSP).  Also, you can use the Options > Display > Axes command1YH6GXP to turn off the plotting of the axes lines and labels in the 2D-plot window.


COPROJECTION(A) approximates to the coprojection of A.  The coprojection of a projection matrix produced by ISOMETRICS is a vector of m+1 two-column matrices that plots as the isometric projection of the same parametric surface, but with lines of constant s instead of constant t.  Superimposing a plot of the coprojection onto a plot of the projection often helps visualize a 3-dimensional surface.  Note that the coprojection can also be generated by approximating ISOMETRICS(v, t, t0, tn, n, s, s0, sm, m).


Note that, unlike parametrically defined surface plots made in a 3D-plot window, hidden lines are not removed from such plots made in a 2D-plot window.


ISOMETRICS and COPROJECTION are often used with functions that generate 3D coordinate vectors like those described in Coordinate System Conversion FunctionsCoordinate_System_Conversion_Functions.  For example, to draw a sphere of radius 2 in a 2D-plot window plot

ISOMETRICS(SPHERE(2,theta,psi),theta,-pi,pi,16,psi,0,pi,8)

and

COPROJECTION(ISOMETRICS(SPHERE(2,theta,psi),theta,-pi,pi,16,psi,0,pi,8))

with the Options > Approximate Before Plotting optionEVZ132 turned on.


Other Graphics FunctionsZG4ZTI 

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