Tutorial: Generation of 3D objects from 2D contours through extrusion and rotation

Prof. Dr. Tim C. Lueth, Professor at Technical University of Munich, MIMED - Department of Mechanical Engineering, Munich, August 2025

clear, close all   % Important for script development. The line can be removed later.

==== PART 1 ============================================================

An example contour that we want to extrude

CPLsample(7); CPL=ans;

When two contours, CPLA and CPLB, are connected to form a body, then in the simplest case there are

A lower surface (floor) consists of the tessellation/triangulation of CPLA (surface orientation facing downwards).
An upper surface (cover) consists of the tessellation/triangulation of CPLA (surface orientation facing upwards).
One wall (outward for the right-hand rule, inward for the left-hand rule)

Converting the embedded CPL into a point list and an edge list with "PLELofCPL"

Wir zerlegen CPLA in PLA und ELA

[PL,EL]=PLELofCPL(CPL)

PL = 22×2

EL = 22×2

SGfigure(0,90); VLELplots(PL,EL,'r.-',.5,.5,.5);

For the ground floor and ceiling cover, we tessellate PLA and ELA into triangles using "delaunayTriangulation."

TR=delaunayTriangulation(PL,EL);

VLFLplot(TR.Points,TR.ConnectivityList,'r',0.1);

However, we are only looking for triangles that are enclosed by the edge list with "isInterior".

We select the facets enclosed by the edges of the contour. "isInterior" selects only the facets that are inside of edges that define a triangle facet by tight hand rule

iia=isInterior(TR);
VLA=TR.Points; FLA=TR.ConnectivityList(iia,:);

We still need to flip the normal vector of the surfaces, as the ground floor of the body must point downwards.

FLA=FLA(:,[1 3 2]); % Change in the direction of the nromal vector of the surface from top to bottom

VLA=VLaddz(VLA,0);

VLFLplotalpha(VLA,FLA,'m',1,'k');

VLFLplot(VLA,FLA,'m',2,5); view(-30,30)

We can now represent the bottom of the body as a surface.

SGfigure(-30,30); VLFLplot(VLA,FLA,'g');

Ceiling cover with the same contour but normal vector pointing upwards

VLB=VLaddz(VLA(:,1:2),30); FLB=TR.ConnectivityList(iia,:);

We can now represent the bottom of the body and the lid as surfaces.

SGfigure(-30,30); VLFLplot(VLB,FLB,'r','',5); VLFLplot(VLA,FLA,'g','',5);

We can automatically determine the wall surfaces of a contour from the edge lists.

nva=size(VLA,1)

nva = 22

VL=[VLA;VLaddz(VLA(:,1:2),30)];

FL1=[EL(:,1) EL(:,2) EL(:,2)+nva]; % FLcontourwallELn

FL2=[EL(:,1) EL(:,2)+nva EL(:,1)+nva]; % FLcontourwallELn

FLWA=[FL1;FL2];

VLFLplot(VL,FLWA,'y','',5);

However, this can only be achieved so easily (with two lines) if the two contours at the top and bottom are similar

have the same number of points
have the same orientation/direction of rotation (clockwise/counterclockwise)
have matching starting points when forming the surface
have the same bend directions
have the same distances between the bends
have the same points on the straight lines between the bends

For embedded contours, the two contours must also

consist of the same number of outer contours
the respective outer contours of A can be assigned to corresponding outer contours of B
this can also be applied to all inner contours and their inner outer contours
missing inner contours can be closed or missing inner outer contours can be started

[VL,FL,FLW,FLU,FLO]=VLFLofPLELz(VLA(:,1:2),EL,30);

SGfigure(-30,30);

VLFLplotalpha(VL,FLW,'y',0.9);

VLFLplotalpha(VL,FLU,'g',1);

VLFLplotalpha(VL,FLO,'r',0.9);

==== PART 2 ============================================================

"SGofCPLz" Erzeugen von Körper durch Extrusion in der Höhe z

SGofCPLz(CPLsample(38),15);

SGofCPLz(CPLsample(38),[-10 10]);

SGofCPLz([PLcircle(15);nan nan;PLcircle(16)],15)

ans = struct with fields:
    VL: [224×3 double]
    FL: [448×3 double]

SGofCPLz([PLcircle(15);nan nan;PLcircle(16)],15,'floor',5); VLFLplotlight(1,0.5);

SGofCPLz([PLcircle(15);nan nan;PLcircle(16)],15,'cover',5); VLFLplotlight(1,0.5);

SGofCPLz(PLsquare(30,20),10);

SGofCPLz(PLsquare(30,20),10,'wall',2,'floor',2);

SGofCPLz(PLsquare(30,20),10,'wall',2,'floor',2,'cover',2); VLFLplotlight(1,0.5);

SGofCPLz(PLsquare(30,20),10,'wall',2,'floor',2,'flsub',PLcircle(7)); VLFLplotlight(1,0.5);

SGofCPLz(PLsquare(30,20),10,'wall',2,'cover',2,'csub',PLcircle(7)); % csub works only without walls

SGofCPLz([PLcircle(15);nan nan;PLcircle(16)],15,'cover',5,'csub',PLsquare(10)); % csub works only without walls

"SGofCPLextrude"

SGofCPLextrude(CPLsample(7),50','y');

SGofCPLextrude(CPLsample(7),50','x');

"SGofCPLflat"

SGofCPLflat(CPLsample(7));

"SGofCPLrot"

SGofCPLrot(PLtransP(CPLsample(7),[30 0]));

"SGofCPLrota"

SGofCPLrota(PLtransP(CPLsample(7),[30 0]),pi);

"SGofCPLzchamfer"

SGofCPLzchamfer(CPLsample(8),10,1);

"SGofCPLzdelaunayGrid" if on edges and in plane x and y should be auxiliary grid points

SGofCPLzdelaunayGrid(CPLsample(8),10,1,1,1);

PLFLofCPLdelaunayGrid: (line 65)

PL = 96×2

8079    1.9509
4346    2.8521
0613    3.7533
6880    4.6545
3147    5.5557
6249    6.2455
9352    6.9352
2455    7.6249
5557    8.3147
6545    8.6880

"SGaddsurfpoints" if also in z there should be auxiliary grid points

SGaddsurfpoints(SGofCPLz(CPLsample(8),10),'',[1,1,1]);

SGmakedoublevertex

ans = struct with fields:
    VL: [2632×3 double]
    FL: [5250×3 double]
    FC: [5245×3 double]

"SGofCPLztwist"

SGofCPLztwist(PLrope([4 1],[1 .5],false),5);

"SGofCPLextrudealongCPL"

SGofCPLextrudealongCPL(PLsquare(3),PLsquare(40)); % Edge Problems - Outdated better use "SGofCPLcontourinstead"

"SGofCPLcontour"

SGofCPLcontour(PLsquare(3),PLsquare(40));

r = 3

SGofCPLcontour(PLsquare(3),PLsquare(40),false);

r = 3

"SGofCPLTL" (uses "SGof2CPLzheurist")

CPLC=PLsquare([60 40]);

CPLC=[CPLC;nan nan;CPLbuffer(CPLC,-5)];

CPLC=CPLsample(33)*2;

PL=[0 0; 0 80; 80 80; 80 40; 160 0];

PL=VLaddz(PL); PL(end,3)=80;

Ro=rofcirclearoundCPL(CPLC);

TL=TLofCVL(PL,Ro*1.1,'','','rad',false);

% SGofCPLTL(CPLC,TL);