ON FRACTAL MODELING OF 3D CURVES AND WIREFRAMES Wieslaw Kotarski (kotarski@gate.math.us.edu.pl) & Agnieszka Lisowska (alisow@ux2.math.us.edu.pl) Institute of Computer Science Silesian University Bedzinska 39 41-200 Sosnowiec, Poland

In our previous Maple applications: "On fractal modeling of contours" http://www.maplesoft.com/applications/app_center_view.aspx?AID=1651and "Probabilistic approach to fractal modeling of shapes" http://www.maplesoft.com/applications/app_center_view.aspx?AID=1657 we showed how to generate fractally any planar contour 2D basing on the Ron Goldman idea [1]. Here we demonstrate fractal construction of non-planar 3D curves that can be splitted into a finite number of cubic Bezier segments. Details concerning theory one can find in [1] and [2]. We recall only that cubic Bezier curve depends on four control points P0,P1,P2,P3 in R3 expressed in homogeneous coordinates. The IFS that generates cubic Bezier curve fractally has the following form: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 on the idea of Ron Goldman [1] it is not possible to construct fractally linear segments, that in fact are Bezier arcs, of the first order. In [2] we showed how to overcome that limitation in 2D case. It appears that linear segments in 3D may be constructed similarly as in 2D case. We start from the well-known fact that the linear segment of lenghts 1 lying on the x -axis joining points [0,0,0] and [1,0,0] can be generated fractally with the usage of the two following matrices: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To obtain any segment joining two points P1=[x1,y1,z1,1] and P2=[x2,y2,z2,1] we need to determine the IFS of the form: 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where 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The M is expressed by S(s), Ry(-beta),Rx(alpha) ,T(x1,y1,z1) that denote scale matrix, rotations matrices along y and x - axises and translation matrix, respectively. Parameters of transformations are the following: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 should be pointed out that we use the left clockwise oriented coordinate system similarly as Maple uses. Below we put the figure that can be useful to understand the idea that lies behind the construction of any linear segment in 3D.TUZOV3RLVWI8b2I8Uj1NRExDZE5ieGo6SDpEVUtcXFZ5Sj08T2BMb1xcanl5eXlRdGZ5c2hDO2lWTENUSmNEWG9YdW12aVlFQUVEP0l4cVdpYVdYTltjZ2FmdD9GPXdTXFxHZjtFSVtLQ0FNYzxxUlxcP2ZGZVI7X2JCeWVaUWJcXFtSS0tlWnFGQl1EcXlCWj93RF1ST09iXFxLY15BSDtdRGE/Y1xcVUNEc0ROc0J0Q2ZPO0ZyU0VFUUlBV3VXa1ZQeVh1V2h1c1dYWVNqP3doXUZ5UXV1QXlHWWVpYWlTd3hgbUhKPVRwQ0U9V3RYX2JQeUVdYHBMPFRnPExebnJqRl47QF5PP2JVP2NaT2M9Plxcdl5uO0haSmllYlhvRj5sWVlmOkBnQGVpZ0c7Z0ZaWUJ0V1NXa2Q6UURSTUJ0Z0U+YUNqUXRaeXlxTXhCR0Jsd0NSO0ZEbXNmWVJIQUJBeXNSO1ZneUNoV2NFX0VKeXZuPUQ+O1hSRUNDV0JgX2RieXVeTHlbbXB1aU47XFxwbkV1b0hTb0VYWHVOUUhsSTxZXFx4cWA8WGxQdldYa0VRWXJ1ckpwV2VRdUh5dT9xTm5lTndUbU91S1g8VnJod0FsS2g9d2xod3RgcEFgVWlVd2VRVWRpcDxJTHl1TENxS1dMWGRsblxcTVhZYXhaREs6VXlZeXVzZE88aFFXXXdhVXFxRXNqeHVodE5MeXZeQVZedXRicU9KaHdQZFBueFVBUG5BYGxqQFQ+UHNxcVk6ZW9qVWtxWFZmYHk6PXdKXVh5cVBXdEtEUVlmUUppPUtUeWpaRE06bU1NSHhabUtkdGtjdE0+XFxZW2xUYzxWYVlsSUlqQkxVT1RvdXVqXl1RbkxOU3FKcGx2XUFqbW10S1hLbUBOdV1uSExwWHVUXFxtTHRUTlp4TEhVb0VBVUJQUEQ8UEVRd0hdc2ZxeGlRWDxlUlZoSnJoU2VBbldETEZoalBUcDtAUnJ1amtATFZUdHQ8bXJwVmxwdXNsT21xb0ZFcXZIeVV0SmJMVWJ2bFpnbHBAdFhAaHNYeUFvW3Jna1BOW1ppXFxoUFpjR3A/bmpeQWpScHQ7Pm5BSWVSZ2lrUV5hX3M7dnNWYWlub25MZ3dzVm87SV9bWHJWeV9Ad1pfXndHZndaUVp3WHZveHJoaWdKSXJHVm9FaHNbWG89P148UXVgZndiQXRAcWBMUXlEVlBFWGFxd3ZrUlFLaTxRR1VbeVBldz5XZUVHQ0FbYkl5REtlZkptRmdhVmlJdTpBdHdnREVzaWxHYnhNY18/c0pRdGNRZWVLYmo/YlZrR0V1RlpNV0htckVneHRzR0dpc3FXZVA7SE07WEVfRWBJQ0Z1YkJRREljd2RVeTtddUI7ZWZHWXlDZW5Lc21Bc1hjckpHaTxjSDtzRk1DUjpTRVRZZmdDd1FJdm47dkBlQ1dbZ29FZj1rd213VD1DdklDSVc7WGNhaGVvd0RzVUJJdzpXRW9FRlk7WXE7ZHlfczpld2NHSW51YmN3Y3dDcnd1d2pzWVdbVmBdRWlBQ0lRd1BTRXlfY0VvdV89YlQ9VEp5eXBXV3hJSWZjYlRVZztDYmRxeFxcT3ZRd3JYc0VsPVV5Y2lBZ3lBeUhGaXJtcWRBO2VnW2lJY0ZoV2JTRWk9UUJEPVhFa3ZheXc8UXU8V2U9YVlSXXJOR0VAcUVEW0JZYWRJTVlrS3U/R2NOT1JCO3ljQ3I+T2hUWXdBPWc/U0hyV2c/QUNxa3hKdUZHQUJYbWRCV0RbZ1RoW1VLYWRYV3hPYUJLS0dpc3lbQ2JiQUNrP1R1P0hZWUZNTWRYdWd3Q3Y8Y3RVRWJxX1hoR0VyT0lOVWVib1RmS1NXVUNvaWJ3QXM/W2ZtbXdvSUlTV2VJUUNhS1d0W1Q/a1ZuSWdiV0lLY0RZQ2dkRUg+b1JiWVdVTUk8S2NDaXlzW2dFQ0lpVVVHUWRvR3lgSWZtO0d3V0NVS0lHWVhmTVVoS1NJRXN0VWlLaWJNZ2N4S0RvPXJTSUZvV0RvV2NUQ0NoV0RtU0NKTXJpYWJpP3JiY2VCU3dPXVhfYWJuR2JWQ0dTVWRwZ0Jac2dqTWJJZ1dFO3VlPWdJY1c6U3NUaUVoUXZoZ3lOUUJqX1RNW3VsP0JDT0dLRUJoP2RgZ1VqP0J2UVRGV2dAU2hEO3NjT3RdW3I/X0dIa2dfO0hfUUdMXWh2SXlIUUJKYWR2cXZFX3VqTVNAO3I/X3l1T0lZPVk+X1NFPUZhcXNRU2h5cVRTd3dJVVM6c0RsVVlbWXI/dVNXQ0NJa3Y6eXRyP2RqP2d0T3JHTVVhO0N1W2VOSUdoW1Y/W2NPU1ZbaXlWR0J4YWVrVXR3Y0lNSUZCQUJNa0JCeVJGO0hNX2tLSVN1QXI7VFRoWEtaQWpCSHRbeXU9YFBfYFVzbVNUYWM6Plo6RmM/b2M+b288P2Y8Mzw=Detailed analysis of the expression for 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shows that because of the special form of matrix F1 the formula for FM1 may be reduced to: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 that means that, in reality, this expression does not depend on scale and rotation matrices and depends only on translation. FM2 may also be simplified similarly as FM1. Further simplifications lead to the following formulas:TUZOV3RLVWI8b2I8Uj1NRExDZE5GWnJbb0daPVp4T3Fid0tuWkpfY05nYnl5eXldT09yTXFJSEpRYU5cXEBOZFxcUWdxWEZ5dXFIV1pMa115d1ZYV11xVEtVd1xcXFxqdlxcbFBpbkZcXGtbVXdKRVRdTFFuXXN5d3h5Pl89TmRBPlxcaE5vSndaQGBrRj5jPnh5OkhqeT5yPGdcXE0+Y1tvYkJHWm8/Z2FOXk5IXkdQdD5YaWFhbWl2c3NYX0RfZEA+ZGZmcEN4bXlweXRpdWl4a1lpWlp5cXJud19xc1VJcUlJcks/Z3JWZlR2XFxMeWtfdnRleGQ7aHJDPnJmaVo6eGVnZ25raXdaSXJQT149P2R4ZlpyP21idnk+aGw7PmVsPmhiRmd1XmtDV2RZdnc7Zm5JVlpLUG9NQVxcWVFraz5yRFB0WT9wckhsQEBmTT95Z3lmdD93VU9cXGhRdE1JaFVZbWNZYlFBaVdIYlpHeHFhX0hIbj93XVJodmZucGJZYkVwbVhQaE9uYWJOWm5RWktJaWFRcmp5anBZaFBPXWxeWkFRbkl2a2RRc3lQWkdHaEh4czpJcGZnZmhJZltGYz0/XkB4XFxkUV9Yb3NfR3ZlSGZJaXJdSGt1eWlkeG1keWpvQWtNbl5gYWBGcVtHd3BuR3Fqb3F2Pm1MSGxQWXliYHJlYGp5TlxceW5pYEluc1lqckd3al5teWBqeU5cXEN3bnZ4YUR5XU1uZW1YbmtnYHdZdlJvXFx3WHNmbnlzPmlSXmRIaWZqQV1bQGNYd2dlZ2t1Tm9ybmJVbl1gSXlNWWJmSWlFT2t5TmVpPnhcXGFyUnl4PnhhTHJ2X0VJR0d5bWZ0VWJ4P3RIbXhGdURwb1VQY2N0cWVRVWdSb0d2RXdGYXlDd0lUa0hOd0lMc0ZYd2hhQ3k9YXl1SWNUR2RTW1c+c1loZ0VQX0R5XUZMT0ZOPVN4QUhia0hfa1VuW1dQTXQ8T1ZvQ3RfQ1RHWXdEbXlqQ1RcXGVldVVmakdVUGNCP3lYPFdDQ1FCSHFYSW1pUHNzQEVJbFN3dD9odHF0TnFVSGV5bTt1Pl90eTt2Y29Ja1VTYG9HZVFocktwPlhyU1RXVUBWTUxZWG1XS1VzXjx5R1hwQXl1WEBUWVxccXlxV05Jd19BVlFcXFBuSFZTbFhqSVZcXHRVWFxcbztcXHVwaXFOQVA7QUxGSXZDXVlCPFRHcGpzWW5bVXQ8SGpYdWw+dVhadVVreWpVPXlNRXBqeVZYUVBxVVdQSHlwWXNcXHh4YFh3d01veXlSb2lqcT1xRE1sS2hvZGlTVl1yWURMRmhTXFx4cTpJc1hRVGpldEFJTUBwVV1wcUZVeDtVd3A9UWJcXFFuSXQ9eVlZXFxUW0lWRHl1d2BYWXFubmF3TWx5PXFSZkRMZmFzXFxNTVZdUVZocVpQWT5gTD1NTmdYdGFITmNsdj5ETVNRUUtZeDplTnVxbV5hcEBod3RJT3hxVlhFc1ZgbHhYVE1ddm5FU3NcXHhTZVdndHM9RXlsTU5naXVLdVVkYW5zcVNSXFxub0BWeXRucURKQXlSVUFPdWlWWlR0XVxcdmR1WF5xbEtAd2U8cUN4S21hTEZJeE1pcE55V1tdUFt5d09sWWtIT1hhcHd5VkE8cnR5a0hAanVYakRQUHRVTFk8UmtYeW5JUWZscHFVb0o8WXlVUVlUcHhRcW9pa1ptdFlAVW5Jcl5Zd1FUa2BQSkhldj9McV08U049Vz1Qd3dJdlxcaVZCdW1HZWpMZHRxZVRjTXRheG9jcW5EZHhicFFQdXhudFJReVdNdUs/YXJnaXZkRVZecXhCbW1ERFFOdXdkbEpqYFlReEs8XW1kWVh4XFxKXWxrYGFrXUV4W2R4XnR0RXhPTXlXXUlLWGF2TGB5U3huPHVwYnBZWXRyP1R5ZkVsdHlQTj1XbVFsb0FZX1l5Y2hKRXFKc1VOcVBZXUlYd3lyd2xZPVRLVVhXbXhTYmlPZko6YG5cXHROXFx0VFs8UDsyOw==from which it follows that IFS for linear segments depends only on its ends i.e. points P1=[x1,y1,z1,1]and P2=[x2,y2,z2,1].The above mentioned observations help us to improve Maple code for generation of any linear segment. Especially, using the above nice tricks one can simplify Maple codes given in [2] and [3]. In this worksheet we demonstrate some examples of construction of 3D curves and wireframes generated fractally using both deterministic and probabilistic approaches.

restart;with(linalg):with(plots):DEFINITION OF HOMOGENEOUS POINT IN 3DP:=(x,y,z)->[x,y,z,1]:TRANSLATION MATRIXT:=(x,y,z)->[[1,0,0,0],[0,1,0,0],[0,0,1,0],[x,y,z,1]]:POINT TRANSFORMATIONTransPoint:=proc(p,t) [t[1]*p[1]+t[2]*p[2]+t[3]*p[3]+t[4],t[5]*p[1]+t[6]*p[2]+t[7]*p[3]+t[8], t[9]*p[1]+t[10]*p[2]+t[11]*p[3]+t[12]] end:FACE TRANSFORMATIONTransFace:=proc(face,t) local i; [seq(TransPoint(face[i],t),i=1 .. nops(face))] end:OBJECT TRANSFORMATIONTransObj:=proc(obj,t) local i; [seq(TransFace(obj[i],t),i=1 .. nops(obj))] end: IFS3d PROCEDUREIFS3d:=proc(obj,n,List_of_Trans) local i, j, k, s, ResObj: ResObj:=obj; for j to n do s:=NULL; for i to nops(List_of_Trans) do s:=s,seq(TransObj(op(k,[ResObj]),List_of_Trans[i]),k=1 .. nops([ResObj])) od; ResObj:=s; od; polygonplot3d([ResObj],axes=boxed,labels=[x,y,z],color=green,scaling=constrained,thickness=5) end:GIVEN INITIAL 3D OBJECT - BOXv1:=[1,0,0]: v2:=[1,1,0]: v3:=[1,1,1]: v4:=[1,0,1]: v5:=[0,0,0]: v6:=[0,1,0]: v7:=[0,1,1]: v8:=[0,0,1]: front:=[v1,v2,v3,v4]: back:=[v6,v5,v8,v7]: left:=[v1,v4,v8,v5]: right:=[v2,v6,v7,v3]: bottom:=[v1,v5,v6,v2]: top:=[v4,v3,v7,v8]: box:=[front,back,left,right,bottom,top]: GIVEN INITIAL 3D OBJECT - TETRAHEDRONtetra:=[[[1., 1., 1.866025404], [1., 1.816496581, .7113248653], [.2928932190, .5917517094, .7113248653]],[[1., 1., 1.866025404], [.2928932190, .5917517094, .7113248653], [1.707106781, .5917517094, .7113248653]],[[1., 1., 1.866025404], [1.707106781, .5917517094, .7113248653], [1., 1.816496581, .7113248653]],[[1., 1.816496581, .7113248653], [1.707106781, .5917517094, .7113248653], [.2928932190, .5917517094, .7113248653]]]:3D BEZIER CURVE AS FRACTALDEFINITION OF BERSTEIN POLYNOMIALSB:=(i,n,u)->(n!/(i!*(n-i)!)*(1-u)^(n-i)*u^i):3D BEZIER CURVEKB3d:=proc(p) local Qx,Qy,Qz,a,b: Qx:=(u,p)->sum('p[i,1]*B(i-1,nops(p)-1,u)','i'=1..nops(p)): Qy:=(u,p)->sum('p[i,2]*B(i-1,nops(p)-1,u)','i'=1..nops(p)): Qz:=(u,p)->sum('p[i,3]*B(i-1,nops(p)-1,u)','i'=1..nops(p)): a:=plot3d([Qx(u,p),Qy(u,p),Qz(u,p)],u=0..1,v=0..1,color=red); b:=PLOT3D(CURVES(p),COLOUR(RGB,0,1,0),LINESTYLE(3)); display(a,thickness=4,axes=boxed,labels=[x,y,z]); end:AN EXAMPLE OF 3D BEZIER CURVEKB3d([[2,1,3],[6,2,4],[1,-3,1],[5,0,-3],[3,4,5],[6,6,2]]);FRACTAL 3D BEZIER CURVE GENERATION PROCEDURE. Four points P1,P2,P3,P4,initial object and n - the number of iterations should be given. GenBez3d:=proc(P0,P1,P2,P3,box,n) local PP,L, M, LP, MP, bezier3d; PP:=[P0,P1,P2,P3]; L:= [[B(0,0,1/2),0,0,0],[B(0,1,1/2),B(1,1,1/2),0,0], [B(0,2,1/2),B(1,2,1/2),B(2,2,1/2),0], [B(0,3,1/2),B(1,3,1/2),B(2,3,1/2),B(3,3,1/2)]]; M:=[[B(0,3,1/2),B(1,3,1/2),B(2,3,1/2),B(3,3,1/2)],[0,B(0,2,1/2),B(1,2,1/2), B(2,2,1/2)],[0,0,B(0,1,1/2),B(1,1,1/2)],[0,0,0,B(0,0,1/2)]]; LP:=evalm(inverse(PP)&*L&*PP); MP:=evalm(inverse(PP)&*M&*PP); bezier3d:=[[LP[1,1],LP[2,1],LP[3,1],LP[4,1],LP[1,2],LP[2,2],LP[3,2],LP[4,2],LP[1,3],LP[2,3],LP[3,3],LP[4,3]],[MP[1,1],MP[2,1],MP[3,1],MP[4,1],MP[1,2],MP[2,2],MP[3,2],MP[4,2],MP[1,3],MP[2,3],MP[3,3],MP[4,3]]]; IFS3d(box,n,bezier3d); end:

AN EXAMPLE - FRACTAL GENERATION OF CUBIC 3D BEZIER CURVEstarting from a boxGenBez3d(P(0,0,0),P(0,1,0),P(1,0,0),P(0,0,1),box,6);starting from a tetrahedron GenBez3d(P(0,0,0),P(0,1,0),P(1,0,0),P(0,0,1),tetra,6);starting from the segment joining points [0,0,0] and [1,1,1]GenBez3d(P(0,0,0),P(0,1,0),P(1,0,0),P(0,0,1),[[[0,0,0],[1,1,1]]],6);ANIMATIONstarting from a boxfor n from 0 by 2 to 8 do GenBez3d(P(0,0,0),P(0,1,0),P(1,0,0),P(0,0,1),box,n); print(`Iteration` = n); od;starting from a tetrahedronfor n from 0 by 2 to 8 do GenBez3d(P(0,0,0),P(0,1,0),P(1,0,0),P(0,0,1),tetra,n); print(`Iteration` = n); od;starting from the segment joining points [0,0,0] and [1,1,1]for n from 0 by 2 to 8 do GenBez3d(P(0,0,0),P(0,1,0),P(1,0,0),P(0,0,1),[[[0,0,0],[1,1,1]]],n); print(`Iteration` = n); od;FURTHER EXAMPLES EXAMPLE 1Using Bezier Proceduredisplay( KB3d([[0,0,0],[0,1,0],[1,0,0],[0,0,1]]), KB3d([[0,0,1],[0,-1,0],[-1,0,0],[0,0,2]]), KB3d([[0,0,2],[0,2,1],[2,0,1],[0,0,3]]), KB3d([[0,0,3],[0,-2,0],[-2,0,0],[0,0,0]]), orientation=[120,30],labels=[x,y,z],scaling=unconstrained);Generated Fractallydisplay( GenBez3d(P(0,0,0),P(0,1,0),P(1,0,0),P(0,0,1),box,8), GenBez3d(P(0,0,1),P(0,-1,0),P(-1,0,0),P(0,0,2),box,8), GenBez3d(P(0,0,2),P(0,2,1),P(2,0,1),P(0,0,3),box,8), GenBez3d(P(0,0,3),P(0,-2,0),P(-2,0,0),P(0,0,0),box,8), orientation=[120,30],labels=[x,y,z],scaling=unconstrained);Animationfor n from 0 by 1 to 8 do display( GenBez3d(P(0,0,0),P(0,1,0),P(1,0,0),P(0,0,1),box,n), GenBez3d(P(0,0,1),P(0,-1,0),P(-1,0,0),P(0,0,2),box,n), GenBez3d(P(0,0,2),P(0,2,1),P(2,0,1),P(0,0,3),box,n), GenBez3d(P(0,0,3),P(0,-2,0),P(-2,0,0),P(0,0,0),box,n), orientation=[120,30],labels=[x,y,z],scaling=unconstrained); print('Iteration'=n); od;EXAMPLE 2Using Bezier Proceduredisplay(KB3d([[0,-1,0],[1,-1,0],[-1,1,0],[0,-1,1]]), KB3d([[0,-1,1],[0,-2,0],[-2,0,0],[0,-1,2]]), KB3d([[0,-1,2],[0,-3,0],[-3,0,0],[0,-1,3]]),orientation=[20,50],labels=[x,y,z], scaling=unconstrained);Generated Fractallydisplay( GenBez3d(P(0,-1,0),P(1,-1,0),P(-1,1,0),P(0,-1,1),box,8), GenBez3d(P(0,-1,1),P(0,-2,0),P(-2,0,0),P(0,-1,2),box,8), GenBez3d(P(0,-1,2),P(0,-3,0),P(-3,0,0),P(0,-1,3),box,8), orientation=[20,50],labels=[x,y,z],scaling=unconstrained);

PROCEDURE FOR 3D SEGMENT GENERATION (two points,starting object and n - the number of iterations are needed)GenSe3d:=proc(P1,P2,obj,n) local F1,F2,segment; F1:=[[1/2,0,0,0],[0,1/2,0,0],[0,0,1/2,0],[0,0,0,1]]; F2:=[[1/2,0,0,0],[0,1/2,0,0],[0,0,1/2,0],[1/2,0,0,1]]; segment:=[[F1[1,1],F1[2,1],F1[3,1],P1[1]/2,F1[1,2],F1[2,2],F1[3,2],P1[2]/2,F1[1,3],F1[2,3],F1[3,3],P1[3]/2], [F2[1,1],F2[2,1],F2[3,1],P2[1]/2,F2[1,2],F2[2,2],F2[3,2],P2[2]/2,F2[1,3],F2[2,3],F2[3,3],P2[3]/2]]; IFS3d(obj,n,segment); end:FRACTAL GENERATION OF WIREFRAME 3D OBJECT - BOXa:=GenSe3d(P(1,1,0),P(1,1,1),tetra,5):b:=GenSe3d(P(0,0,0),P(0,0,1),tetra,5):c:=GenSe3d(P(0,0,0),P(0,1,0),tetra,5):d:=GenSe3d(P(0,1,0),P(0,1,1),tetra,5):e:=GenSe3d(P(0,0,1),P(0,1,1),tetra,5):f:=GenSe3d(P(1,0,0),P(1,0,1),tetra,5):g:=GenSe3d(P(1,0,0),P(1,1,0),tetra,5):h:=GenSe3d(P(1,0,1),P(1,1,1),tetra,5):i:=GenSe3d(P(0,1,0),P(1,1,0),tetra,5):j:=GenSe3d(P(0,1,1),P(1,1,1),tetra,5):k:=GenSe3d(P(0,0,0),P(1,0,0),tetra,5):l:=GenSe3d(P(0,0,1),P(1,0,1),tetra,5):display(a,b,c,d,e,f,g,h,i,j,k,l);obj:=box:ANIMATION starting from a boxdisplay(seq(display(GenSe3d(P(1,1,0),P(1,1,1),obj,n),GenSe3d(P(0,0,0),P(0,0,1),obj,n), GenSe3d(P(0,0,0),P(0,1,0),obj,n),GenSe3d(P(0,1,0),P(0,1,1),obj,n),GenSe3d(P(0,0,1),P(0,1,1),obj,n),GenSe3d(P(1,0,0),P(1,0,1),obj,n),GenSe3d(P(1,0,0),P(1,1,0),obj,n),GenSe3d(P(1,0,1),P(1,1,1),obj,n),GenSe3d(P(0,1,0),P(1,1,0),obj,n),GenSe3d(P(0,1,1),P(1,1,1),obj,n),GenSe3d(P(0,0,0),P(1,0,0),obj,n),GenSe3d(P(0,0,1),P(1,0,1),obj,n)),n=0..7),insequence=true,scaling=constrained); seg:=[[[1,1,1],[2,2,2]]]:FRACTAL GENERATION OF WIREFRAME 3D OBJECT - TETRAHEDRONdisplay( GenSe3d(P(0,0,0),P(.3,1,.5),seg,8), GenSe3d(P(0,0,0),P(1,.3,.5),seg,8), GenSe3d(P(0,0,0),P(1,1,.3),seg,8), GenSe3d(P(.3,1,.5),P(1,.3,.5),seg,8), GenSe3d(P(1,.3,.5),P(1,1,.3),seg,8), GenSe3d(P(.3,1,.5),P(1,1,.3),seg,8), orientation=[120,75]);ANIMATION starting from a tetrahedrondisplay(seq(display( GenSe3d(P(0,0,0),P(.3,1,.5),tetra,n), GenSe3d(P(0,0,0),P(1,.3,.5),tetra,n), GenSe3d(P(0,0,0),P(1,1,.3),tetra,n), GenSe3d(P(.3,1,.5),P(1,.3,.5),tetra,n), GenSe3d(P(1,.3,.5),P(1,1,.3),tetra,n), GenSe3d(P(.3,1,.5),P(1,1,.3),tetra,n), orientation=[120,75]),n=0..6),insequence=true,scaling=constrained);ANIMATION starting from a segmentdisplay(seq(display(GenSe3d(P(0,0,0),P(1,1,1),[[[1,2,3],[3,4,3]]],n),GenSe3d(P(1,1,1),P(2,3,4),[[[1,2,3],[3,4,3]]],n),GenSe3d(P(0,0,0),P(2,3,4),[[[1,2,3],[3,4,3]]],n),labels=[x,y,z],orientation=[-120,70]),n=0..8),insequence=true,scaling=constrained);

PROCEDURE FOR CREATING TWO FUNCTIONS f1,f2 FROM GIVEN COEFFICENTS: a,b,c,d,e,f,g,h,i,j,k,lMakeMapFromCoeffs:=proc(a,b,c,d,e,f,g,h,i,j,k,l) local x, y, z; unapply([a*x+b*y+c*z+j,d*x+e*y+f*z+k,g*x+h*y+i*z+l],x,y,z); end: MakeMapFromCoeffs(1,2,3,4,5,6,7,8,9,10,11,12):GENERATOR OF UNIFORMLY DISTRIBUTED NUMBERS 1 or 2 d:=rand(1..2): PROBABILISTIC PROCEDURE FOR GENERATING OF QUADRATIC BEZIER CURVE.Four non-colinear points P0,P1,P2,P3 and n - the number of iterations should be given).Starting point is chosen as P0 i.e. it belongs to the attractor! BezProb3d:=proc(P0,P1,P2,P3,n) local L, M, PP, LP, MP, i, l, z, dd, x, y, f1, f2, zz: PP:=[P0,P1,P2,P3]; L:= [[1,0,0,0],[1/2,1/2,0,0],[1/4,1/2,1/4,0],[1/8,3/8,3/8,1/8]]; M:=[[1/8,3/8,3/8,1/8],[0,1/4,1/2,1/4],[0,0,1/2,1/2],[0,0,0,1]]: LP:=evalm(inverse(PP)&*L&*PP); MP:=evalm(inverse(PP)&*M&*PP); f1:=MakeMapFromCoeffs(LP[1,1],LP[2,1],LP[3,1],LP[1,2],LP[2,2],LP[3,2],LP[1,3],LP[2,3],LP[3,3],LP[4,1],LP[4,2],LP[4,3]); f2:=MakeMapFromCoeffs(MP[1,1],MP[2,1],MP[3,1],MP[1,2],MP[2,2],MP[3,2],MP[1,3],MP[2,3],MP[3,3],MP[4,1],MP[4,2],MP[4,3]); x[0]:=P0[1]: y[0]:=P0[2]: z[0]:=P0[3]: zz[0]:=[x[0],y[0],z[0]]: for i from 0 to n do dd:=d(): if dd=1 then zz[i+1]:=evalf(f1(zz[i][1],zz[i][2],zz[i][3])): else zz[i+1]:=evalf(f2(zz[i][1],zz[i][2],zz[i][3])): fi: od: i:='i': l:={seq(zz[i],i=0..n)}: pointplot3d(l,symbol=CROSS,axes=none,color=red,axes=boxed,labels=[x,y,z]): end: AN EXAMPLEBezProb3d(P(0,0,0),P(0,1,0),P(1,0,0),P(0,0,1),3000);ANIMATIONfor n from 0 by 50 to 500 do display( BezProb3d(P(0,0,0),P(0,1,0),P(1,0,0),P(0,0,1),n), BezProb3d(P(0,0,1),P(0,-1,0),P(-1,0,0),P(0,0,2),n), BezProb3d(P(0,0,2),P(0,2,1),P(2,0,1),P(0,0,3),n), BezProb3d(P(0,0,3),P(0,-2,0),P(-2,0,0),P(0,0,0),n), orientation=[120,30],labels=[x,y,z],scaling=unconstrained); print('Iteration'=n); od;display( BezProb3d(P(0,-1,0),P(1,-1,0),P(-1,1,0),P(0,-1,1),1000), BezProb3d(P(0,-1,1),P(0,-2,0),P(-2,0,0),P(0,-1,2),1000), BezProb3d(P(0,-1,2),P(0,-3,0),P(-3,0,0),P(0,-1,3),1000), orientation=[20,50],labels=[x,y,z],scaling=unconstrained);THE SECOND GENERATOR OF UNIFORMLY DISTRIBUTED NUMBERS 1 or 2 d1:=rand(1..2):SEGMENT GENERATED FRACTALLYSegProb3d:=proc(P1,P2,n) local F1, F2, f1, f2, zz, x, y, z, l, dd,i; F1:=[[1/2,0,0,0],[0,1/2,0,0],[0,0,1/2,0],[0,0,0,1]]; F2:=[[1/2,0,0,0],[0,1/2,0,0],[0,0,1/2,0],[0,0,0,1]]; f1:=MakeMapFromCoeffs(F1[1,1],F1[2,1],F1[3,1],F1[1,2],F1[2,2],F1[3,2],F1[1,3],F1[2,3],F1[3,3],P1[1]/2,P1[2]/2,P1[3]/2); f2:=MakeMapFromCoeffs(F2[1,1],F2[2,1],F2[3,1],F2[1,2],F2[2,2],F2[3,2],F2[1,3],F2[2,3],F2[3,3],P2[1]/2,P2[2]/2,P2[3]/2); x[0]:=P1[1]: y[0]:=P1[2]: z[0]:=P1[3]: zz[0]:=[x[0],y[0],z[0]]: for i from 0 to n do dd:=d1(): if dd=1 then zz[i+1]:=evalf(f1(zz[i][1],zz[i][2],zz[i][3])): else zz[i+1]:=evalf(f2(zz[i][1],zz[i][2],zz[i][3])): fi: od: i:='i': l:={seq(zz[i],i=0..n)}: pointplot3d(l,symbol=CROSS,color=red,axes=boxed,labels=[x,y,z]): end: SegProb3d(P(0,0,1),P(1,2,3),1000);ANIMATION - TETRAHEDRON GENETATED FRACTALLYdisplay(seq(display( SegProb3d(P(0,0,0),P(.3,1,.5),n), SegProb3d(P(0,0,0),P(1,.3,.5),n), SegProb3d(P(0,0,0),P(1,1,.3),n), SegProb3d(P(.3,1,.5),P(1,.3,.5),n), SegProb3d(P(1,.3,.5),P(1,1,.3),n), SegProb3d(P(.3,1,.5),P(1,1,.3),n), orientation=[120,75]),n=[0,2,5,10,20,50,100,200,300,500]),insequence=true,scaling=constrained);

Here in this worksheet we demonstrated some examples of fractal rendering of 3D curves. More complex 3D curves that can be splitted exactly or approximately into a sum of cubic 3D Bezier arcs and linear segments may be generated fractally using the methodology presented in this Maple application. It should be pointed out that fractal rendering is a progressive one. Further it should be mentioned that it seems to be possible to render fractally also Bezier patches [1]. But that problem is still an open one.

[1] Goldman R.; The fractal nature of B\351zier curves, Proceedings of the Geometric Modeling and Processing 2004, April 13-15, Beijing, China, 2004, 3-11. [2] Kotarski W., Lisowska, A. On fractal modeling of contours, Maplesoft, 2005, http://www.maplesoft.com/applications/app_center_view.aspx?AID=1651. [3] Kotarski W., Lisowska, A. Probabilistic approach to fractal modeling of shapes, Maplesoft, 2005, http://www.maplesoft.com/applications/app_center_view.aspx?AID=1657. The undersigned (the "Author") has developed an Application (the "Application") using Maple and wishes to submit the Application to Maplesoft, a division of Waterloo Maple Inc. for publication by Maplesoft or its agents. 1. The Author grants to Maplesoft a non-exclusive, perpetual License and right to use all or any part of the Application, to modify or adapt the Application for the promotion of Maplesoft. 2. The Author represents that the Application is the original work of the Author, the Author has the right to grant a License to Maplesoft with respect to the Application, no other party has any right, title or interest in or to the Application, and that the use of the Application by Maplesoft will not infringe any intellectual property rights of any third party. 3. The Author agrees to execute such documents and take such other action as Maplesoft may request in order to give effect to the intention of this License Agreement. This License Agreement shall be binding upon and inure to the benefit of the Author, and all heirs, executors, administrators, successors and permitted assigns of the Author. Wieslaw Kotarski and Agnieszka LisowskaSosnowiec, July 2005