Interactive Graphics Using Parametric Equations Day 2 Dr' Niels Lobo Computer Science

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Interactive Graphics Using Parametric Equations
(Day 2)Dr. Niels LoboComputer Science
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Bezier Curves

• Google bezier curves
• http//www.doc.ic.ac.uk/dfg/AndysSplineTutorial/B
  eziers.html/

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Question 1 and choices

• Pierre Bezier was a
• A. automechanic
• mathematician in car design industry
  
• hairdresser
• race car driver

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Interactive Graphics Curves

• Uses
• -- Design of Fonts and other printer symbols
• -- Consumer goods shapes of cell phones, cars,
  etc.

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Bezier Curves

• Curve is specified by 2 equations
• and are curve
  endpoints
• and are
  guidepoints

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A Bezier Curve

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Bezier Curves

• Mathematically, we verified that
• Slope of a handle is same as tangent at endpoint
• i.e., the tangent at has same
  slope as the
• segment joining to

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Graphics Curve in General Form

• Curve is specified by 2 equations, one is
• Which can be re-written as

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General Form of Bezier Curve

• This equation
• 
  can be re-written as

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General Form of Bezier Curve

• This form
• can be re-written as

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General Form of Bezier Curve

• Hence, the equations are (using n3, for 4
  points)
• Generic notation,
• Or,

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General Form of Bezier Curve

• Can be re-written as,
• Or,
• where the organizer can choose n, and the user
• then supplies n1 points.

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General Form of Bezier Curve

• is,
• Most common is n3, some use n2, and n4.
• Abandon notion of handles, and use notion of
• guidepoints.

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General Form of Bezier Curve

• Most common is n3, some use n2, and n4.

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Question 2 and choices

• In this Figure, value of n for the first and
  last is
• A. 3 and 3
• 3 and 5
• 2 and 4
• 4 and 4

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Bezier Curve in Matrix Notation

• Consider the familiar case
• It expands to

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Towards Matrix Notation

• Gives

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Towards Matrix Notation

• Can be written as

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Matrix notation

• Suppose we have 3 Apples 5 Bananas
  10
• 6 Apples 7
  Bananas 15
• What is cost of Apple? cost of Banana?
• Can write as 3A 5B 10
• 6A 7B 15
• In matrix notation, get

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Matrix notation

• Using matrix notation,
• then would solve this Matrix system
• for
• by finding inverse matrix of
• This is studied in class on
• Matrix Algebra or Linear Algebra.

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Matrix Notation

• Matrices are also very useful in Computer
  Graphics
• for dealing with rotations and 3-dimensional
  projections
• For now, we only care about the notation i.e.,
  that we
• can write in one form (English) or the other
  (matrix).
• 
  
• 
  
• 3 Apples 5 Bananas 10
• 6 Apples 7 Bananas 15

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Question 3 and choices

• Write in Matrix notation 13 Cats 15
  Dogs 100
• 
  9 Cats 6 Dogs 65
• A.
  B.
• C.
  D.

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Question 4 and choices

• 11 Cats 8 Dogs 80
  
  6 Cats 5 Dogs 45
• Which is NOT equivalent to above??
• A.
  B.
• C.
  D.

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Back To Bezier Curves

• We had
• Wrote as

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Back To Bezier Curves

• This
• has same form for y
• Generic equation

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Recall the Polynomial Notation

• For curve,
• For surface,

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Bezier Surfaces

• We have
• Or, for mn3

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Bezier Surfaces

• We need
• Figure adapted from Princeton Web site

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Some Properties of Bezier Surfaces

• Four corners are like Tent anchors, i.e., they
• are tied down to fixed points.
  (Interpolation)
• Along any border, the surface behaves as a
• single Bezier curve.
• Just as with single curve, the surface fits in
  the
• Convex Hull of the specified points

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Some Properties of Bezier Surfaces

• Because the four borders are explicit Bezier
  curves,
• they can be linked to neighboring patches,
  by
• making the common border the same Bezier
  curve,
• i.e., the same four control points.

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Demo Bezier Surface

• Google Bezier surface demo
• http//

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Verify that (t0,s0) is tied down

• We have
• i.e.,

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Question 5 and choices

• Given
  what is ?
• A. B. C.
  D.

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Verify

• So have

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Question 6 and choices

• A. B.
  C. D.

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Verify

• So have

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Question 7 and choices

• For
• 
  what is
• 0 0 0 1 B. 0 0 0
  -1
• C. 1 0 0 0 D. -1 3
  -3 1

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Question 8 and choices

• So have
• Which is
• A. B. C.
  D.

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Verify

• So have
• So done verification!!

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Question 9 and choices

• Can join patches because
• They would need to have common 4 points
• They would look good regardless of seam
• They would provide a jump in smoothness
• Not possible to join patches

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Question 10 and Choices