Cynical Surface

7/28/2019 Cynical Surface

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Dr. Regalla Srinivasa Prakash

I/C, Regalla Srinivasa Prakash 1



•

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


smoo ness



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




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




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




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


u u



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.



Contd.

• To create a surface the use be ins bconstructing wireframe entities andconnectin them a ro riatel b ro e

surface entities




• 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


v



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




-)]([][ == 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 +×+




Parametric representation of surface

maxminmaxmin ,

,,,,,

vvvuuu

vu zvu yvu x z y xvu

≤≤≤≤==




Concept of surface patches




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.




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


∂∂∂ vvv



• 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


⎦⎣ ⎠⎝ ∂⎝ ∂⎝ ∂⎝ ∂11100100 PPPP

vvvv



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


v

Cynical Surface

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