(Aktu Btech) Computer Graphics Important Unit-3 Three Dimensional

Achieve B.Tech success with Aktu’s Quantum Notes. Discover the world of Computer Graphics by reading these important notes that include important and recurrent questions. Exam excellence made simple! Unit-3 Three Dimensional

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Important Questions For Computer Graphics:
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Q1. What do you mean by geometric primitives ? Discuss.

Ans. The fundamental geometrical shapes that go into creating computer graphics scenes and the finished visuals are known as geometric primitives.

Commonly used 3D geometric primitives include:

• 1. Points: A point is an exact position or location on a plane surface. 
• 2 Line: Lines are employed to represent linear objects with insignificant width and height when geometry is used to model the real world. 
• 3. Line segments: A line segment is a section of a line that has two end points and includes all of the points on the line in between them.
• 4. Planes: A plane is the two dimensional analogue of a point (zero dimensions), a line (one-dimension) and a space (three-dimensions). 
• 5. Circles: A circle is a straightforward shape in Euclidean geometry made up of points on a plane that are equally spaced apart from a central point. 
• 6. Ellipses: Ellipses, which emerge from the intersection of a circular cone with a plane that does not pass through its apex, are closed curves and the bounded case of conic sections.
• 7. Triangles: A triangle is a polygon with three line-segmented corners, or vertices, and three sides, or edges.
• 8. Polygons: Traditionally, a polygon is a plane figure with a closed path or circuit that is made up of a finite number of straight line segments.
• 9. Spline: A spline is a unique function that is better than polynomial interpolation and is defined piecewise by polynomials.
• 10. Spheres: A sphere, which has the form of a round ball, is a fully spherical geometrical object in three dimensions.

Q2. Write the steps involved in constructing 3D viewing transformation.

Ans. The steps involved in constructing 3D viewing transformations: 

The complete three dimensional viewing process (without hidden surface removal) is described by the following steps:

• 1. Transform coordinates from world coordinates to normalized viewing coordinates by applying the transformation N_par Or N_per.
• 2. Clip in normalized viewing coordinates against the canonical clipping volumes. 
• 3. Project onto the screen projection plane using the projections Par or Per. 
• 4. Apply the appropriate (two dimensional) viewing transformation. 

In terms of transformations, we can describe the above process in terms of a viewing transformation V_T where

Here CL and V₂ refer to the appropriate clipping operations and two dimensional viewing transformations.

Q3. What do you mean by projection ? Differentiate between parallel projection and perspective projection.

Ans. Projection: 

• 1. A projection is a two-dimensional representation of a three-dimensional item or scene; it is the shadow of the object.
• 2. Projections convert coordinate system points of dimension ‘n’ into coordinate system points of dimension ‘n’ less.

Types of projections : 

• 1. Parallel projection: 
  • a. Parallel lines are used to translate coordinate points to the view plane in a parallel projection.
  • b. These are linear transforms that can be used to create scale drawings of three-dimensional objects for use in blueprints.
• 2 Perspective projection: 
  • a. A perspective projection transforms the locations of objects to the view plane via lines that converge at the centre of the projection.
  • b. Calculating the point at which the projection lines and the view plane intersect provide the projected view of an object.

Difference:

Q4. Explain the matrix for perspective projection for three vanishing points.

Ans. Matrix for perspective projection for three vanishing point : 

• 1. A three vanishing point projection occurs when the three principal axes cross the projection plane, meaning that none of the axes is perpendicular to the projection plane.
• 2. To recreate the outline of a 3D object, three vanishing point perspective transformations are required.
• 3. The matrix representation of three vanishing point perspective transformation is

• 4. It has three center of projections, one on x-axis at [-1/p, 0,0,1], second on y-axis at [0, – 1/q, 0, 1] and third on z-axis at [0,0,-1/r,1].
• 5. It also has three vanishing points, one on X-axis [1/p, 0, 0, 1], one on y-axis at [0, 1/q, 0, 1] and third on z-axis at [0, 0, 1/r, 1|.

Q5. What do you mean by 3D clipping ? Discuss in detail.

Ans.

• 1. Extending techniques for two-dimensional clipping can be used to create three-dimensional clipping.
• 2. With 3D clipping, the objects are clipped against the view volume’s boundary planes.
• 3. We would need to test the relative location of the line using the boundary plane equations for the view volume in order to clip a line segment against it.
• 4. By modifying the plane equation of each boundary to include the line endpoint coordinates, which establishes whether the endpoint is inside or outside that boundary.
• a. An endpoint (x, y, z) of a line segment is outside a boundary plane if Ax + By + Cz + D > 0
• b. Similarly, the point is inside the boundary if Ax + By + Cz +D < 0.
• c. Lines with both endpoints outside a boundary plane are discarded, and those with both endpoints inside all boundary planes are saved.
• 5. The intersection of a line with a boundary is found using the line equations along with the plane equation.
• 6. Intersection coordinates (x₁, y₁, z₁) are values that are on the line and that satisfy the plane equation Ax₁+ By₁ + Cz₁ +D = 0, where A, B, C, and D are the plane parameters for that boundary.

Q6. Find the transformation for (a) cavalier with 𝛳 = 45° and (b) cabinet projection with 𝛳 = 30°.

Ans. a. A cavalier projection is an oblique projection where there is no foreshortening of line perpendicular to the xy plane. We then see for f = 1, with 𝛳 = 45°, we have 

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