CAD/CAM /CIM (ME-704) - Important Questions

Explain 2D and 3D geometric transformations used in CAD. Derive the matrix forms for 2D translation, scaling and rotation and the homogeneous 3D transformation matrix for translation, scaling and rotation about the coordinate axes. Show with an example how a point $P(1,2,3)$ is transformed by a translation of $\left(2, -1, 3\right)$ followed by a rotation of $30^{\circ}$ about the Z-axis and a uniform scaling by $1.5$.

What are homogeneous coordinates? Explain why homogeneous coordinates are used in computer graphics and CAD. Derive the 3D homogeneous transformation matrix for a rotation about an arbitrary axis passing through the origin defined by a unit vector $\mathbf{u}=\left(u_x, u_y, u_z\right)$. State Rodrigues' rotation formula and show its matrix form.

Compare and contrast wireframe modelling, surface modelling and solid modelling. Explain Boundary Representation (B-rep) and Constructive Solid Geometry (CSG) approaches for solid modelling. Give advantages and disadvantages of B-rep and CSG with respect to rendering, editing and boolean operations.

Explain cubic Bezier curves. Derive the cubic Bezier curve equation using Bernstein polynomials. State and explain important properties of Bezier curves such as endpoint interpolation, convex hull property and variation diminishing property. For control points $P_0,P_1,P_2,P_3$, write the curve equation and compute the point on the curve for $t=0.25$ symbolically.

Explain B-spline curves and NURBS. Define knot vector, control points and basis functions. Write the Cox–de Boor recursive formula for the B-spline basis functions and explain how NURBS extend B-splines to represent conics exactly. Describe the effect of knots (uniform, open-uniform, non-uniform) on curve shape.

For a cubic Bezier curve with control points $P_0=(0,0),\;P_1=(1,2),\;P_2=(3,2),\;P_3=(4,0)$ compute the coordinates of the point on the curve at $t=0.5$. Show all calculations using the Bernstein polynomial form.

Describe surface generation methods: extrusion, revolution, sweeping and lofting. Explain with diagrams (labelled) the procedure of generating a surface by sweeping a profile along a spatial curve and state typical CAD applications of sweep and loft operations.

Explain Boolean operations in solid modelling (union, intersection and difference). Describe algorithmic approaches used to implement Boolean operations on Boundary Representation models and discuss common numerical and topological issues encountered during Boolean operations.

What is feature-based modelling in CAD? Explain the concepts of machining features and design features. Describe a basic approach for feature recognition from a solid model and discuss its importance in downstream applications such as CAM and CAE.

Describe the IGES and STEP data exchange standards. Explain the main differences between IGES and STEP in terms of data representation, capabilities for transferring solid models, and typical usage scenarios. Mention limitations of IGES that STEP addresses.

A CAD feature is to be transformed by the following sequence: scale by $0.5$ about the origin, rotate by $45^{\circ}$ about the Z-axis, and then translate by $\left(3, -2, 1\right)$. Derive the composite homogeneous transformation matrix $\mathbf{T}_{\text{composite}}$ and show how it is applied to a point given in homogeneous coordinates $\left[x, y, z, 1\right]^T$.

Explain tessellation and triangulation of parametric surfaces for visualization and CAM. Describe criteria used to control tessellation density (such as chordal deviation and normal deviation) and discuss trade-offs between accuracy and computational cost.