object triangle

Once you have opened a Maxima session, load package **draw** and then set global variable **draw_renderer** to **vtk**:

load("draw") $ draw_renderer: 'vtk $

A pyramid constructed by three triangles of different colors:

draw3d( color = blue, triangle([0,0,0], [2,0,0], [3,1,2]), color = yellow, triangle([0,0,0], [2,2,0], [3,1,2]), color = gray, triangle([2,0,0], [2,2,0], [3,1,2]) ) $

Quadrilateral as two adjacent triangles:

draw3d( triangle([0,0,2], [5,0,2], [2,3,2]), triangle([6,3,2], [5,0,2], [2,3,2]), color = red, triangle([0,0,5], [5,0,5], [2,3,5]), triangle([6,3,5], [5,0,5], [2,3,5]) ) $

We now generalize some ideas used above in order to extend draw capabilities related to polygons. First, we need to know how to plot polyhedran angles:

/* v: the vertex of the polyhedran angle, [p]: endpoints of the segments, all of them starting at point v. */ polyhedral_angle(v, [p]) := block([np], if listp(v) and length(v) = 3 and (np: length(p)) > 1 then makelist(triangle(v, p[k], p[k-1]), k, 2, np) else funmake('polyhedral_angle,cons(v,p))) $ /* an example; note that point [1,2,3] is repeated to close the polyhedran angle */ draw3d( color = yellow, polyhedral_angle( [0,0,6], /* <-- the first point is the vertex */ [1,2,3], [2,2,-2], [4,-1,0], [2,-2,2], [-1,-1,2],[-6,-2,3], [-8,5,5], [1,2,3]) )$

If all points are coplanar, we can plot arbitrary convex polygons. In this example, they all belong to the xy plane

draw3d( color = yellow, polyhedral_angle( [0,0,0], [1,2,0], [2,2,0], [4,-1,0], [2,-2,0], [-1,-1,0],[-6,-2,0], [-8,5,0], [1,2,0]) )$

We can now draw regular polygons in all positions and orientations. Let's first define a function that returns a transformation matrix to rotate a point an angle **th** around a line passing through point **poi** and parallel to vector **vec**. After that, we define a function to calculate the vertices of a regular polygon

rotation3d(th,vec,poi) := block([c,s,t,u,v,w,a,b,d], c : cos(th), s : sin(th), t : 1 - c, [u,v,w] : vec / sqrt(apply("+", vec^2)), [a,b,d] : poi, matrix( [u^2+(v^2+w^2)*c, t*u*v-s*w, t*u*w+s*v, (a*(v^2+w^2)- u*(b*v+d*w))*t+(b*w-d*v)*s], [t*u*v+s*w, v^2+(u^2+w^2)*c, t*v*w-s*u, (b*(u^2+w^2)- v*(a*u+d*w))*t+(d*u-a*w)*s], [t*u*w-s*v, t*v*w+s*u, w^2+(u^2+v^2)*c, (d*(u^2+v^2)- w*(a*u+b*v))*t+(a*v-b*u)*s], [0, 0, 0, 1] ))$ /* Generates the vertices of a regular polygon: n: number of vertices u: a vector orthogonal to the polygon p: point origin of vector u v: coordinates of one of the vertices of the regular polygon */ regular_polygon_vertices(n,u,p,v) := block([vs: [endcons(1,v)], m], m: rotation3d(2*%pi / n, u, p), for k:2 thru n do vs: cons(first(transpose(m . first(vs))), vs) , map(lambda([z], [z[1], z[2], z[3]]), vs) )$ /* Here is an example; the coordinates of the penthagon centered at [4,-2,1], with one vertex at [5,2,6/7], and orthogonal to vector [5,2,8] */ points: float(regular_polygon_vertices(5,[5,2,8],[4,-2,1],[5,2,6/7])) $ /* note that we make use of function polyhedral_angle, where the vertex of the polyhedral angle is one of the vertices of the polygon */ draw3d( axis_3d = false, color = gray, apply(polyhedral_angle,points))$

Pressing the **w** key will show the wire structure of the polygon.

Geometric transformation of a triangle. The three vertices are transformed, and then a new triangle is drawn:

poly3: triangle([0,0,0], [2,0,0], [3,1,2]) $ draw3d( poly3, color = red, transform = [x+1,x*y^2,z+1,x,y,z], poly3 ) $

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