Surface models

CAD/CAM Subject-Mechanical engineering

1. ComputerGraphics
Surface Models
•Asurfaceofanobjectismorecompleteandless
ambiguousrepresentationthanitswireframemodel;itis
anextensionofawireframemodelwithadditional
information.
•Awireframemodelcanbeextractedfromasurfacemodel
bydeletingallsurfaceentities(notthewireframeentities–
point,lines,orcurves!).Databasesofsurfacemodelsare
centralizedandassociative,manipulationofsurface
entitiesinoneviewisautomaticallyreflectedintheother
views.Surfacemodelscanbeshadedandrepresentedwith
hiddenlines.

2. ComputerGraphics
Types of Surfaces
1. Plane Surface
•Thisisthesimplestsurface,requires3non-coincidentalpoints
todefineaninfiniteplane.Theplanesurfacecanbeusedto
generatecrosssectionalviewsbyintersectingasurfaceor
solidmodelwithit.

3. ComputerGraphics
2. Ruled (lofted) Surface
Thisisalinearsurface.Itinterpolateslinearlybetween
twoboundarycurvesthatdefinethesurface.
Boundarycurvescanbeanywireframeentity.The
surfaceisidealtorepresentsurfacesthatdonothave
anytwistsorkinks.

4. ComputerGraphics
3.Surface of Revolution
•Thisisanaxisymmetricsurfacethatcanmodel
axisymmetricobjects.Itisgeneratedbyrotatinga
planarwireframeentityinspaceabouttheaxisof
symmetryofagivenangle.

5. ComputerGraphics
4.Tabulated Surface
Thisisasurfacegeneratedbytranslatinga
planarcurveagivendistancealonga
specifieddirection.Theplaneofthecurve
isperpendiculartotheaxisofthegenerated
cylinder.

6. ComputerGraphics
5. Bi-linear Surface
This3-Dsurfaceisgeneratedbyinterpolationof4
endpoints.Bi-linearsurfacesareveryusefulin
finiteelementanalysis.Amechanicalstructureis
discretizedintoelements,whicharegeneratedby
interpolating4nodepointstoforma2-Dsolid
element.

7. ComputerGraphics
6.Coons Patch
•Coons patch or surface is generated by the
interpolation of 4 edge curves as shown.

8. ComputerGraphics
Bi-cubicpatches (Surfaces)
•The concept of parametriccurves can be
extended to surfaces
•The cubic parametric curve is in the form of
Q(t)=t
T
Mqwhere q=(q1,q2,q3,q4) : qi control
points, Mis the basis matrix (Hermite or
Bezier,…), t
T
=(t
3
, t
2
, t, 1)

9. ComputerGraphics
•Now we assume qito vary along a parameter s,
•Qi(s,t)=t
T
M[q1(s),q2(s),q3(s),q4(s)]
•qi(s)are themselves cubic curves, we can write
them in the form …

10. ComputerGraphics
BicubicpatchessMMt
MsMsMttsQ
TT
TTT
....
])[..],...,[...(.),(
444411111
q
q,q,q,qq,q,q,q
4324321


where q is a 4x4 matrix
Each column contains the control points of
q1(s),…,q4(s)
x,y,z computed by











44342414
43332313
42322212
41312111
qqqq
qqqq
qqqq
qqqq sMMttsz
sMMttsy
sMMttsx
T
z
T
T
y
T
T
x
T
....),(
....),(
....),(
q
q
q




11. ComputerGraphics
14/10/2008 Lecture 6 11
Bézier example
•We compute (x,y,z) by coordszofarrayisq
sMqMttsz
coordsyofarrayisq
sMqMttsy
coordsxofarrayisq
sMqMttsx
z
T
BzB
T
y
T
ByB
T
x
T
BxB
T
44
....),(
44
....),(
44
....),(







12. ComputerGraphics
14/10/2008 Lecture 6 12
Continuity of Bicubic patches.
•Hermite and Bézier patches
–C
0 continuity by sharing 4
control points between
patches.
–C
1 continuity when both sets
of control points either side of
the edge are collinear with the
edge.