The document discusses the representation of curves and surfaces in computer graphics, highlighting their importance in modeling real-world objects and applications such as CAD and data plots. It describes various types of curves, including implicit, explicit, and parametric curves, particularly focusing on parametric cubic curves like Hermite, Bézier, and splines. The text emphasizes the need for smooth and stable representations while evaluating the trade-offs between different polynomial degrees used in curve representation.

1. SINTHIA SARKER
Dept. of CSE
GREEN UNIVERSITY OF BANGLADESH
7/21/20201

2. o Curve and Surface
o Parametric cubic curves
o Types
o Mathematical complexity
7/21/20202

3. Curve:
In computer graphics, we often need to draw
different types of objects onto the screen.
Objects are not flat all the time and we need to
draw curves many times to draw an object.
Types:
Implicit curves
Explicit curves
Parametric curves
7/21/20203

4. Surface :
Objects are represented as a collection of
surfaces. Most common representation for
surfaces:
•Polygon mesh
•Parametric surfaces
•Quadric surfaces
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5. There are many ways to represent curves
and surfaces . We want a representation that is
- Stable
- Smooth
- Easy to evaluate
7/21/20205

6. We need smooth curves and surfaces in
many applications:
- Model real world objects
- Computer-aided design (CAD)
- High quality fonts
- Data plots
- Artists sketches
7/21/20206

7. Polylines and polygons:
• Large amounts of data to achieve good accuracy.
• Interactive manipulation of the data is tedious.
Higher-order curves:
•More compact (use less storage).
•Easier to manipulate interactively.
Possible representations of curves:
•explicit, implicit, and parametric
7/21/20207

8. There are Three Types of Parametric Cubic Curves.
Hermite Curves:
Defined by two endpoints and two endpoint tangent vectors
(used 1st order)
Bézier Curves:
Defined by two endpoints and two control points which
control the endpoint’ tangent vectors
Splines:
Defined by four control points
7/21/20208

9. General form:
c= coefficient matrix
T= parameter matrix
GMTCTtztytxtQ
tttT
ddd
ccc
bbb
aaa
C
dtctbtatz
dtctbtaty
dtctbtatx
zyx
zyx
zyx
zyx
zzzz
yyyy
xxxx




















)]()()([)(
]1[
)(
)(
)(
23
23
23
23
7/21/20209

10. GMTCTtztytxtQ  )]()()([)(
If cubic the derivation….
7/21/202010

11. Rewrite the coefficient matrix as
where M is a 44 basis matrix, G is called the
geometry matrix .
So,
where is called the
blending function
7/21/202011

12. • Why cubic?
– lower-degree polynomials give too little flexibility in
controlling the shape of the curve
– higher-degree polynomials can introduce unwanted
wiggles and require more computation
– lowest degree that allows specification of endpoints
and their derivatives
– lowest degree that is not planar in 3D
7/21/202012

13. 7/21/202013