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7/28/2019 Cynical Surface
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Dr. Regalla Srinivasa Prakash
I/C, Regalla Srinivasa Prakash 1
7/28/2019 Cynical Surface
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•
model realistic physical surfaces such as, .
• rea ng sur aces requres somequantitative data (such as points, tangent
vec ors an some qua a ve a a sucas intuition of the desired shape and
I/C, Regalla Srinivasa Prakash 2
smoo ness
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Contd.
• ust like curves surfaces can also beinterpolating as well as approximating
• Bezier surface is an approximator whereas-
• In addtion to modeling geometric objects,surfaces can be used to fit experimental
data, tables of numbers, and discretizedsolution of differential equations
I/C, Regalla Srinivasa Prakash 3
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Contd.
• So the general objective here is to model amulti-dimensional surface, given points in 3-Dspace, and parameter value at each point
• For example:–
distribution over a 2-D domain, z=p(x,y), on an
oblique airplane wing
– Stress distribution in a mechanical part
– Contou lot is an alternative but is normall not clea
to everyone
I/C, Regalla Srinivasa Prakash 4
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WHY WE NEED A PARAMETRIC EQUATION FOR SURFACE?
• Merel locatin some oints on thesurface can also be a very simple butineffective wa of constructin a surface
because the properties of the surface cannotbe ud ed fromit
,approximating function with proper
the properties of the curve far better
I/C, Regalla Srinivasa Prakash 5
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Practice of CAD packages• All CAD packages build surfaces on the
curve description that we learnt till now
• When two curves are specified a ruledsurface may result whereas when only
curve specified a surface may bepro uce y revovng aroun an axs
• The display of a surface is done over amesh of parameters m×n, controlled by
I/C, Regalla Srinivasa Prakash 6
u u
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Surface models
• A surface model is a more complete and lessambiguous representation than its wireframemo e .
• Surface models define only the geometry of their.
• ey s ore no n orma on a ou e opo ogy othese objects.
, ,neither the surfaces nor the edge (curve) itself
I/C, Regalla Srinivasa Prakash 7parametric representations.
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Contd.
• To create a surface the use be ins bconstructing wireframe entities andconnectin them a ro riatel b ro e
surface entities
I/C, Regalla Srinivasa Prakash 8
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• Wrongly digitized locations result in absurdor undesirable surface patches
• Mesh size for dis la affects the
visualization; but higher the mesh sizelon er the CPU time taken for dis la
solids, not in wireframe models
• sa va age o sur ace represen a on sthat the ambiguity is still not completely
I/C, Regalla Srinivasa Prakash 9
v
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Surface entities
• Ruled surface•
• Tabulated cylinder (or surface)• Bicubic Hermite spline surface• Bezier surface - Rectangular and triangular
Bezier patches-• Coons patches• Gordon surface• Fillet surface• Offset surface
I/C, Regalla Srinivasa Prakash 10
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-)]([][ == T x f y xT z y xP
ol nomialaisointdataiventheallthrou hassto
surfaceafor y)f(x,of formnaturalmostThe
=
below.shownas
p qnm
P(x,y,z)
sizeof grid XYan bydescribed issurfacethewhere
,0 0= =m n
mn
po n s.q p +×+
I/C, Regalla Srinivasa Prakash 11
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Parametric representation of surface
maxminmaxmin ,
,,,,,
vvvuuu
vu zvu yvu x z y xvu
≤≤≤≤==
I/C, Regalla Srinivasa Prakash 12
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Concept of surface patches
I/C, Regalla Srinivasa Prakash 13
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I/C, Regalla Srinivasa Prakash 14
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Surface Analysis –Tangent vectors of surface patch
For example, a
topologically
rec angu ar
patch mayhave two
tan ent
vectors
defined at any
point on it.
I/C, Regalla Srinivasa Prakash 15
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obtained issurfaceon thev)P(u, pointanyatector tangent vTheTangent Vectors
=
point.eachfor ectorstangent vtwoarethereThereforeother.theto
≤≤≤≤∂∂+
∂∂+
∂∂=
∂∂= vvvuuu
u
z
u
y
u
x
u
PvuPu ,ˆˆˆ),(
,
maxminmaxmin,
≤≤≤≤∂+∂+∂=∂=
=
vvvuuu z y xPvuP
u
ˆˆˆ
be,it willcurveconstantthealongand
k∂∂∂∂ vvvv
form,matrixIn
j
⎥⎥
⎢⎢
∂∂∂∂∂∂=⎥
⎦
⎤⎢⎣
⎡ z y xuuu
P
P
v
u
I/C, Regalla Srinivasa Prakash 16
∂∂∂ vvv
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• Based on the nature of the surface at any point(ui, v j), the tangent vectors in “u” and “v”
,P
and P
⎟ ⎞
⎜⎛ ∂
⎟ ⎞
⎜⎛ ∂
,
at that point can be at any acute angle to eachijij
PP
.• When they are perpendicular to each other, then
their dotproduct is zero.• The tangent vectors at the fou corners of the
rectangular surface patch are
⎥⎥⎥⎥
⎢⎢⎢⎢
⎟ ⎞
⎜⎛ ∂
⎟ ⎞
⎜⎛ ∂
⎟ ⎞
⎜⎛ ∂
⎟ ⎞
⎜⎛ ∂
⎟⎟ ⎠⎜⎜⎝ ∂⎟⎟ ⎠⎜⎜⎝ ∂⎟⎟ ⎠⎜⎜⎝ ∂⎟⎟ ⎠⎜⎜⎝ ∂11100100 PPPP
PPPP
uuuu
I/C, Regalla Srinivasa Prakash 17
⎦⎣ ⎠⎝ ∂⎝ ∂⎝ ∂⎝ ∂11100100 PPPP
vvvv
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MAGNITUDES OF TANGENT VECTORS AND UNIT TANGENT VECTORS
:ectorstan ent vma nitudesThe
u
z y xP
P⎟
⎞⎜⎛ ∂
+⎟ ⎞
⎜⎛ ∂
+⎟ ⎞
⎜⎛ ∂
==∂
22 2
z y xP
uuuu
⎛ ∂⎛ ∂⎛ ∂∂22 2
vvvvv ⎠⎝ ∂ ⎠⎝ ∂ ⎠⎝ ∂∂
rsunit vectotheand
uu
Pn =ˆ
v
u
Pn =ˆ
I/C, Regalla Srinivasa Prakash 18
v