📘 Ellipse and Other Curves — Exam Notes
🔹 1. Introduction
In computer graphics, curves are used to create smooth and complex shapes such as circles, ellipses, and free-form designs.
👉 After lines and circles, ellipse and curves are essential for drawing realistic objects.
🔹 2. Ellipse
✔️ Definition
An ellipse is a closed curve where the sum of distances from two fixed points (foci) is constant.
✔️ Equation of an Ellipse
For center ((x_c, y_c)):
[
\frac{(x - x_c)^2}{a^2} + \frac{(y - y_c)^2}{b^2} = 1
]
Where:
- (a) = semi-major axis
- (b) = semi-minor axis
✔️ Special Case
- If (a = b), the ellipse becomes a circle
✔️ Symmetry of Ellipse
- Ellipse has 4-way symmetry:
[
(x, y), (-x, y), (x, -y), (-x, -y)
]
👉 Only one quadrant needs to be calculated.
🔹 3. Midpoint Ellipse Algorithm (Important)
✔️ Concept
- Similar to midpoint circle algorithm
- Uses decision parameter
- Divides ellipse into two regions
✔️ Regions of Ellipse
-
Region 1 (Slope > -1)
-
Region 2 (Slope < -1)
✔️ Initial Conditions
[
x = 0,\quad y = b
]
✔️ Decision Parameters
Region 1:
[
p_1 = b^2 - a^2b + \frac{a^2}{4}
]
Region 2:
[
p_2 = b^2(x + 0.5)^2 + a^2(y - 1)^2 - a^2b^2
]
✔️ Steps (Simplified)
Region 1:
-
Start from (0, b)
-
If (p_1 < 0):
-
Else:
- Move diagonally → (x = x + 1, y = y - 1)
-
Update decision parameter
Region 2:
-
Continue until (y = 0)
-
If (p_2 > 0):
-
Else:
- Move diagonally → (x = x + 1, y = y - 1)
✔️ Advantages
- Efficient
- Uses integer arithmetic
- Accurate
🔹 4. Other Important Curves
🔸 4.1 Parabola
✔️ Definition
A parabola is a curve where each point is equidistant from a focus and a directrix.
✔️ Equation
[
y^2 = 4ax \quad \text{or} \quad x^2 = 4ay
]
✔️ Applications
- Satellite dishes
- Reflectors
🔸 4.2 Hyperbola
✔️ Definition
A hyperbola is a curve formed by two separate branches.
✔️ Equation
[
\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1
]
✔️ Features
- Two open curves
- Symmetric about axes
🔸 4.3 Bezier Curves (Very Important)
✔️ Definition
A Bezier Curve is a smooth curve defined by control points.
✔️ Key Idea
- Curve does not necessarily pass through all control points
- Used for smooth design
✔️ Formula (General)
[
P(t) = \sum_{i=0}^{n} B_{i,n}(t) \cdot P_i
]
Where:
- (P_i) = control points
- (B_{i,n}(t)) = Bernstein polynomials
- (0 \le t \le 1)
✔️ Simple Example (Quadratic Bezier)
3 control points:
- Start point
- Control point
- End point
✔️ Applications
- Fonts
- Animation
- Graphic design
🔸 4.4 B-Spline Curves
✔️ Definition
A B-Spline curve is a generalization of Bezier curves with better control.
✔️ Features
- Smooth curves
- Local control (changing one point affects only part of curve)
🔹 5. Comparison of Curves
| Curve Type |
Shape |
Usage |
| Circle |
Closed |
Wheels, shapes |
| Ellipse |
Closed |
Orbits, designs |
| Parabola |
Open |
Reflectors |
| Hyperbola |
Open |
Scientific models |
| Bezier |
Smooth |
Design, animation |
| B-Spline |
Smooth |
Advanced modeling |
🔹 6. Diagram Descriptions
✔️ Ellipse
- Draw oval shape
- Label axes (a and b)
✔️ Parabola
- U-shaped curve
- Show focus and directrix
✔️ Bezier Curve
- Draw 3–4 control points
- Show curve passing near them
🔹 7. Important Terms
- Control Points: Points that guide curve shape
- Focus: Fixed point used in curve definition
- Directrix: Reference line
- Parametric Curve: Defined using parameter (t)
📝 Likely Exam Questions
- Define ellipse and write its equation.
- Explain midpoint ellipse algorithm.
- What are the regions in ellipse drawing?
- Define parabola and hyperbola with equations.
- Explain Bezier curves and their properties.
- Differentiate between Bezier and B-Spline curves.
- What is symmetry in ellipse drawing?
- Write short notes on different curves.
⚡ Quick Revision Summary
-
Ellipse equation:
[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
]
-
Uses 4-way symmetry
-
Midpoint ellipse algorithm has 2 regions
-
Parabola: (y^2 = 4ax)
-
Hyperbola: (x^2/a^2 - y^2/b^2 = 1)
-
Bezier curves use control points
-
B-Spline gives smoother and flexible curves