CHAPTER 4 &5 geoand trans.pptxGeometry chapter 4 and 5 transformation ,translation ,animation,openGl graphics Geometry chapter 4 and 5 transformation ,translation ,animation,openGl graphics

CHAPTER 4
GEOMETRY AND
LINE GENERATION

1. Introduction to Points and Lines Points
A point is a fundamental element in geometry, representing
a location in space.
In a 2D coordinate system, a point is defined by its
coordinates (x,y)(x,y).
Lines
 A line is a straight one-dimensional figure having no
thickness and extending infinitely in both directions.
 In computer graphics, a line is often represented by its
endpoints (x1,y1)(x1
,y1
) and (x2,y2)(x2
,y2
).

Line Drawing Algorithms
Digital Differential Analyzer (DDA): A simple
algorithm to draw lines by incrementing xx or yy by a
small step.
Bresenham’s Line Algorithm: An efficient
algorithm that uses integer arithmetic to draw
lines, avoiding floating-point calculations.

2. Bresenham’s Line Algorithm
Key Idea
 Bresenham’s algorithm determines the best pixel to plot
at each step to approximate a straight line between two
points.
 It uses decision parameters to choose between two
possible pixels at each step.

Advantages of Bresenham’s Algorithm:
 Works using only integer calculations (addition and
subtraction), making it faster than floating-point
operations.
 Avoids multiplication and division, ensuring efficiency.
 Produces a visually smooth and accurate line.

Generating Circles:
Concept:
 Circles are common geometric shapes that need to be
rendered in computer graphics.
 Efficient algorithms exist to generate circles on raster
displays.
Midpoint Circle Algorithm:
 Concept: An algorithm that uses the midpoint property of a
circle to determine the closest pixel to the true circle. It
leverages the eight-way symmetry of a circle.
 How it works: The algorithm calculates pixels for one octant
of the circle, and the other pixels are found by symmetry.
 Equation of a circle: (x - h)^2 + (y - k)^2 = r^2, where (h,
k) is the center and r is the radius.

Real-life application:
 GUI design: Drawing circular buttons, icons, and other UI
elements.
 Medical imaging: Generating circular regions of interest in
scans.
 Computer graphics: Creating circular shapes in various
applications.

4.2. Plotting General Curves
Concept:
 Beyond straight lines, many applications require the generation of
various curves.
 Curves can be represented mathematically in different ways.
 Parametric Curves:
 Concept:
 A curve where the x and y coordinates are defined as functions of a
parameter, usually denoted as 't'.
 Representation: x = f(t), y = g(t).
 Example:
 A spiral can be defined parametrically: x = t * cos(t), y = t * sin(t).
 Real-life application:
 Animation: Creating complex motion paths for characters and objects.
 CAD: Designing smooth and intricate curves for product design.
 Robotics: Defining the path of a robot arm.

Bézier Curves:
Concept:
 Curves defined by a set of control points. The curve passes
through the first and last control points, but not necessarily the
intermediate ones.
 The control points influence the shape of the curve.
Example:
 A quadratic Bézier curve is defined by three control points.
 A cubic Bézier curve is defined by four control points.
 Font design: Designing smooth and scalable font characters.
 Vector graphics: Used in programs like Adobe Illustrator to
create and manipulate curves.
 3D modeling: Creating smooth surfaces for objects.

Spline Curves:
Concept:
 Piecewise polynomial curves that provide flexibility and
smoothness.
 B-splines are a type of spline that offers local control, meaning
changing one control point only affects a portion of the curve.
Example:
 A B-spline can be used to model the smooth surface of a car body.
 CAD: Surface modeling and design.
 Computer animation: Creating smooth and natural-looking
movements.
 Industrial design: Designing products with complex curved
surfaces.

4.3. Line Thickness
Concept:
 The ability to vary the thickness of a line.
 Thicker lines can be used to emphasize certain features or improve
visibility.
Example:
 Drawing a thick line to represent a major highway on a map, and a thin
line to represent a minor road.
 Technical drawings: Conveying different levels of importance or
different materials.
 Mapping: Distinguishing between different types of roads or boundaries.
 User interface design: Creating visually distinct borders for elements.
Techniques:
 Drawing multiple parallel lines.
 Using a thicker pixel mask or brush.
 Approximating with a filled rectangle.

4.4. Line Style
Concept:
 The appearance of a line can be modified by using different styles, such
as dashed, dotted, or a combination of dashes and dots.
 Different line styles can convey different meanings or improve visual
clarity.
Example:
 Using a dashed line to represent a hidden object or a boundary that is
not physically present.
 Technical drawings: Representing hidden lines, center lines, or
phantom lines.
 Map design: Differentiating between different types of boundaries or
features.
 Computer networks: Showing different types of network connections.
Techniques:
 Using a repeating pattern (mask) of pixels to create the desired style.

4.5. Polygons
Concept:
 A polygon is a closed shape formed by a sequence of connected line
segments.
 Polygons are fundamental building blocks for representing 2D shapes.
Representation:
 Vertex list: A list of the coordinates of the polygon's vertices, in order.
 Example:
 A triangle, a rectangle, and a hexagon are all examples of polygons.
 2D graphics: Representing shapes in games, illustrations, and user
interfaces.
 CAD: Modeling the shapes of objects and parts.
 GIS: Representing areas such as buildings, parks, and administrative
regions on maps.

 Polygon types:
 Convex: All interior angles are less than 180 degrees.
 Concave: At least one interior angle is greater than 180 degrees.
 Simple: Edges do not intersect each other.
 Complex: Edges intersect each other.

4.6. Filling
 Concept:
 The process of coloring the interior of a polygon or a bounded region.
 Filling is essential for rendering solid shapes and creating visually complete
images.
 Scan-Line Algorithm:
 Concept: Fills a polygon by processing it one horizontal line (scan line) at
a time.
 How it works: The algorithm determines which pixels on each scan line
are inside the polygon and fills them with the desired color or pattern.
 Example: Filling a square or a circle on a computer screen.
 Rendering filled shapes in 2D graphics.
 Image editing software.
 Drawing applications.
 Example: Filling an irregularly shaped area in an image with a solid color.
 Image editing (e.g., the "paint bucket" tool).
 Interactive graphics applications.
 Game development.

Boundary-Fill Algorithm:
Concept: Fills a region that is enclosed by a
boundary of a specific color.
How it works: Starts from a point inside the
region and fills until it hits a pixel of the
boundary color.
Example: Filling the inside of a drawn circle,
where the circle outline is black.
Real-life application:
Image editing.
Graphics software.

4.7. Text and Characters
 Concept:
 The representation and display of text in computer graphics.
 Text is essential for providing information, labels, and user interfaces.
 Character representation:
 Bitmap fonts: Characters are stored as pixel patterns (arrays of bits).
 Vector fonts: Characters are defined by mathematical outlines (curves
and lines).
 Example:
 Displaying a sentence on a screen using a specific font and size.
 User interfaces: Displaying menus, buttons, and labels.
 Word processing: Rendering text in documents.
 Web browsers: Displaying text on web pages.
 Video games: Showing dialogue, scores, and other information.

 Text rendering techniques:
 Rasterization: Converting vector font outlines into pixel
patterns for display on a raster device.
 Anti-aliasing: Techniques for smoothing the jagged edges
of text characters to improve readability.
 Text attributes:
 Font: The typeface or style of the text (e.g., Arial, Times
New Roman).
 Size: The height of the characters.
 Style: Variations such as bold, italic, and underline.
 Color: The color of the text.
 Alignment: How the text is positioned within a given area
(e.g., left-aligned, centered).

output
 Explanation:
 display():
 This function contains the drawing commands.
 glClear(GL_COLOR_BUFFER_BIT) clears the screen.
 glColor3f() sets the drawing color.
 glBegin() and glEnd() define the shapes to be drawn.
 glVertex2f() defines the coordinates of the vertices.
 The house is drawn using GL_QUADS for the body, door, and window, and
GL_TRIANGLES for the roof.
 reshape():
 This function handles window resizing and sets up the orthographic projection.
 main():
 This function initializes GLUT, creates the window, sets the background color,
and starts the main loop.
 Key Improvements:
QUIZ (10%)
1. Exact output 6%
2. Modification 8-10pt(it
dependence)

CHAPTER 5
GEOMETRICAL TRANSFORMATIONS
Introduction:
3D geometrical transformations are
fundamental operations that manipulate the
position, orientation, and size of 3D objects
in a virtual space. They are essential in
computer graphics, animation, robotics,
virtual reality, and many other fields.

5.1 3D Transformation
3D transformations manipulate objects in three-dimensional
space (x, y, z).
Common 3D transformations include:
 Translation: Moving an object in 3D space.
 Scaling: Changing an object's size along x, y, or z-axis.
 Rotation: Rotating an object around an axis.
 Reflection: Mirroring an object over a plane.
 Shearing: Skewing an object along an axis.

Translation:
A translation transform simply moves every point by a certain
amount horizontally and a certain amount vertically. If (x,y, z ) is
the original point and (x1,y1 z1) is the transformed point, then the
formula for a translation is
Real-life application: Robot arm movement, moving
a point cloud in 3D scanning.

Rotation:
Rotating an object around a specified axis (x, y, or z).
Example: Rotating a satellite to point its antenna.
Real-life application: Flight simulator controls, rotating
surgical instruments in virtual surgery

Scaling:
Changing the size of an object along the x, y, and z axes.
Example: Resizing a 3D model of a building.
Real life application: 3D printing of objects at varying sizes,
adjusting the size of a virtual object in a game.

Shearing:
Distorting the shape of an object by shifting its
coordinates along one axis relative to another.
Example: Creating a slanted appearance for a building.
Real life application: Used in image processing to correct
distortions, creating special effects in animation.

Reflection:
Mirroring an object across a plane.
Example: Creating a reflection of a
building in water.
Real life application: Simulating reflections
in virtual environments, creating
symmetrical designs in CAD.

OpenGl code that demonstrate
rotation, translation, scaling, reflection
and shearing using codeblock

CHAPTER 4 &5 geoand trans.pptxGeometry chapter 4 and 5 transformation ,translation ,animation,openGl graphics Geometry chapter 4 and 5 transformation ,translation ,animation,openGl graphics

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