undefined | unit 2 curves surfaces

SMD

Unit 2

Curves and Surfaces

Q1) Explain the non-parametric representation of curves

A1) Non-parametric or Cartesian representation

For a nonparametric curve, the coordinates y and z of a point on the curve are expressed as two separate functions of the third coordinate x as the independent variable.

In other words, in non-parametric or cartesian system, the curve is represented as a relation between the coordinates x, y, z.

There are two forms of non-parametric representation of curves:

Explicit representation

In explicit representation, coordinates are expressed as a function of any one independent coordinate.

Explicit Non-parametric representation for 2D curve is

Explicit Non-parametric representation for 3D curve is

b. Implicit representation

In implicit representation, the curve is represented as a relation between the coordinates.

Implicit Non-parametric representation for 2D curve is

 Limitation of non-parametric or cartesian or generic representation of curves:

Due to one-to-one relation between the coordinates x, y and z of a point, the explicit representation cannot be used for a closed curves like circles.

The implicit representation of curves requires solving of simultaneous equations, which is highly inconvenient and lengthy.

Also in non-parametric representation of the curves, the equation depends on the coordinate system used.

If the slope of a curve at a point is vertical or near vertical, its value becomes infinity or very large.

Shapes of most engineering objects are intrinsically independent of any coordinate system.

If the curve is to be displayed as a series of point or straight-line segments, the computations involved could be extensive.

Q2) Explain Parametric Representation of Curves

A2)

All the difficulties in non-parametric representation are overcome in parametric representation.

In parametric form, each point on a curve is expressed as a function of a parameter u.

Here, the coordinates are not in relation with each other but are the function of an independent parameter u [f(u)].

This independent coordinate acts as a local coordinate for the points on the curve.

Parametric representation for 2D curve is

Where,

Parametric representation for 2D curve in matrix form is

Where,

The tangent vector at point P is

Parametric representation for 3D curve is

Where,

Parametric representation for 2D curve in matrix form is

Where,

The tangent vector at point P is

Q3) Explain C0, C1 and C2 continuity

A3) If each section of a spline is described with a set of parametric coordinate functions of the form

Where,

Zero Order Continuity C0

Zero-order parametric continuity, described as C0 continuity, means simply that the curves meet.

The end points of the two curve segments meet with each other in C0 continuity.

Zero-order continuity C0 yields a position continuous curve.

That is, the values of x, y, and z evaluated at u, for the first curve section are equal, respectively, to the values of x, y, and z evaluated at u, for the next curve section.

Mathematically,

(x, y, z for curve segment 1 at u = umax) = (x, y, z for curve segment 2 at u = umin)

2. First order continuity C1

First-order parametric continuity, C1 continuity, means that the first parametric derivatives (tangent lines) of the coordinate functions for two successive curve sections are equal at their joining point.

C1 continuity imply slope continuous curves

At the joining of two segments, the tangent or first order derivative of parametric equation of both curve segment coincides or are equal.

Mathematically,

(x’, y’, z’ for curve segment 1 at u = umax) = (x’, y’, z’ for curve segment 2 at u = umin)

3. Second Order Continuity C2

Second-order parametric continuity, or C2 continuity, means that both the first and second parametric derivatives of the two curve sections are the same at the intersection.

C2-order continuity imply curvature continuous curves.

At the joining of two segments, first order derivative as well as second order derivative of parametric equation of both curve segment coincides or are equal.

Mathematically,

(x’’, y’’, z’’ for curve segment 1 at u = umax) = (x’’, y’’, z’’ for curve segment 2 at u = umin)

Q4) Explain need of synthetic curve and approaches to generate a synthetic curve

A4) Need of Synthetic curve

The need for synthetic curves in design arises on two occasions: when a curve is represented by a collection of measured data points and when an existing curve must change to meet new design requirements.

Analytic curves are usually not sufficient to meet geometric design requirements of mechanical parts. Synthetic curves provide designers with greater flexibility and control of a curve shape by changing the positions of the control points.

Products such as car bodies, ship hulls, airplane fuselage and wings, propeller blades, shoe insoles, and bottles are a few examples that require free-form, or synthetic, curves and surfaces.

A spline curve is defined by giving a set of coordinate positions, call control points, which indicate the general shape of the curve. These control points are then fitted with piecewise continuous parametric polynomial functions.

When polynomial sections are fitted so that the curve passes through each control point, the resulting curve is said to interpolate the set of control points.

On the other hand, when the polynomials are fitted to the general control point path without necessarily passing through any control point, the resulting curve is said to approximate the set of control points.

There are two approaches to generate synthetic curve

Interpolation

In interpolation, the curve passes through all the data points.

Approximation

In approximation, curve does not pass through all data points but are close to data points.

Q5) Explain Bezier curve and give its characteristics

A5) “The Bezier curves uses the given data points for generating the curve and passes through first and last data points while other acts as control points.”

Bezier curves are used more than hermite cubic splines because of its flexibility to change shape of curve.

Bezier curve

Parametric equation of Bezier curve is written as,

Where

Bezier curve for n+1 data point is nth degree polynomial.

Characteristics of Bezier curve

Does not use tangent vectors but data points for controlling its shape

Bezier curve for n+1 data point is defines by nth degree polynomial.

The sequence of the data points can be reversed without changing the shape of the curve.

A closed Bezier curve can be generated by closing its polygon. i.e. End points are made coincident

The flexibility of Bezier curve increases by increasing data points.

Q6) Give properties of NURBS curve

A6) Properties of NURBS curve

NURBS curve passes through the first and the last control points if nonperiodic knots are used.

The tangent vector at the starting point is in the same direction as P1-P0 and at the ending point it is in the same directions as Pn - Pn-1

NURBS curve equation is a general form that can represent both B-spline and NURBS curves. A Bezier curve is considered to be a special case of a B-spline curve, so the NURBS equation can also represent Bezier and rational Bezier curves.

The shape of a B-spline curve is modified by changing the x, y, and z coordinates of the control points; that is, three degrees of freedom are allowed for each control point. However, the homogeneous coordinates hi in addition to the x, y, and z coordinates of each control point can be changed in a NURBS curve. Thus, a more versatile modification of a curve becomes possible if the curve is represented by a NURBS equation.

Increasing the value of the homogeneous coordinate of a control point has the effect of drawing a curve toward the control point

Q7) Derive the expression for Coons patch surface

A7) The corner points are blended in a bilinear surface, but four boundary curves are blended to form a surface in a Coon's patch. The word patch is used to indicate explicitly that the surface being generated is a surface segment corresponding to the parameter region . Thus, any arbitrary surface is composed by many of these patches.

The equation of a Coon's patch is derived as follows. We assume that the equations of the four boundary curves are given by P0(v), P1(v), 'Q0(u), and Q1(u), as illustrated in Figure.

We also assume that the curves 'lo(u) and Q1(u) have the interval of u from 0 to 1 and the same direction. In Figure, we assume that their direction is to the right, as indicated by the arrow for u. Similarly, P0(v) and P1(v) have the interval of v from 0 to 1 and the same direction, upward in this case.

First, two curves facing each other are selected, say, P0(v) and P1 (v). Then they are interpolated in the u direction by a linear equation:

Now let's try to define another surface by interpolating Q0(u) and Q1(u) in the v direction:

Now, let's try yet another surface, P3(u, v), defined by adding P1(u, v) and P2(u, v) to determine whether it can be bounded by all the boundary curves. Then P3(u, v) becomes

and substituting the limit values of u and v

Each underlined term is the linear interpolation between the end points of the corresponding boundary curve. In other words, the terms to be eliminated are the expressions of the boundary curves of a bilinear surface

Therefore, the correct equation of a Coon's patch is obtained by subtracting the bilinear surface equation from P 3(u, v):

Q8) Write a note on Reverse Engineering

A8) Reverse engineering is the process of obtaining a geometric CAD model from measurements acquired by scanning an existing physical model. The measurements are in the form of 3D point clouds that correspond to points on the surface of the object being re-engineered. Using CAD models to represent the scanned object is very important in various industries because they help improve the quality and efficiency of design. In addition, they speed up the manufacturing and analysis process.

Reverse Engineering Methodology

The characterized typical procedure of reverse engineering shown in figure.

Reverse Engineering: It consists of five steps: (1) data acquisition, (2) preprocessing (noise filtering and merging), (3) triangulation, (4) feature extraction, and (5) segmentation and surface fitting. Data acquisition and processing systems includes hardware and software components. A hardware system acquires point clouds or volumetric data by using available experimental setup. A software system processes raw point clouds or volumetric data and transfers them into a virtual representation of object surfaces. The point cloud data is acquired in the form of x, y and z co-ordinates of the multiple point of the object surface. The scanning techniques use to scan the object are contact and noncontact technique.3D digitization system such as non-contact 3D scanner generated the large amount of data. In general, scanning data can be saved in different file formats, out of which the point cloud and STL formats are very useful for research assessment.

Q9) Write equation of line having end points P1(3,5,8) and P2(6,4,3). Find the tangent vector and coordinates of points on line at u=0.25,0.5,0.75.

A9)

P1(3,5,8) P2(6,4,3)

Parametric equation of ine:

This is the equation of line.

Tangent vector of line:

Coordinates of point:

At u = 0.25

At u = 0.5

At u = 0.75

The coordinates of points at

u = 0.25 is R(3.75,4.75,6.75)

u = 0.5 is S(4.5,4.5,5.5)

u = 0.75 is T(5.25,4.25,4.25)

Q10) A circle is represented by centre point (5,5) and radius 6 units. Find the parametric equation of circle and determine the various points on the circle in first quadrant if increment of angle is 45o and 90o.

A10)

Pc (xc, yc, zc) = (5,5,0)

Parametric equation of circle is given by

Where,

Coordinates of points on circle are given in table:

Points	u	x	y	(x,y)
P1	0	11	5	(11,5)
P2	45	4.5	9.24	(4.5,9.24)
P3	90	5	11	(5,11)

Q11)Calculate the points on Hermite cubic spline curve at u = 0, 0.2, 0.4, 0.6, 0.8 having end points P0 (4,4) and P1 (8,5). The tangent vector for ends P0 and P1 defined by line between P0 and P2 (5,6) and line between P1 and P3 (10,7).

A11)

P0 (4,4) P1 (8,5) P2 (5,6) P3 (10,7)

Equation for x-coordinate:

P0x = 4 P1x = 8 P2x = 5 P3x = 10

Slope of tangent is given by

P’0x = P2x – P0x = 5 – 4 = 1

P’1x = P3x – P1x = 10 – 8 = 2

parametric equation of hermite curve for x-coordinate passing through two points P0 and P1 and two tangent vectors at these points P0’ and P1’ respectively is

Equation for y-coordinate:

P0y = 4 P1y = 5 P2y = 6 P3y = 7

Slope of tangent is given by

P’0y = P2y – P0y = 6 – 4 = 2

P’1y = P3y – P1y = 7 – 5 = 2

General parametric equation of hermite curve passing through two points P0 and P1 and two tangent vectors at these points P0’ and P1’ respectively is

The parametric equation for hermite cubic spline is

Points on the hermite cubic spline are given in table

u	0	0.2	0.4	0.6	0.8	1
Px(u)	4	4.48	5.36	6.4	7.36	8
Py(u)	4	4.3	4.45	4.55	4.7	5
(x,y)	(4,4)	(4.48,4.3)	(5.36,4.45)	(6.4,4.55)	(7.36,4.7)	(8,5)

Sign Up