Manufacturing Technology (ME461)

Instructor: Shantanu Bhattacharya

Review of Previous Lecture

Limitations of a wireframe model.

Non-parametric representation of equation (Implicit case).

Parametric equation based representation of an ellipse.

Parametric representation of synthetic curves.

Concepts of continuity.

Hermite cubic spline fit.

Example Problem

Determine and plot the equation of Hermite form of a cubic spline form given position vectors and slopes at the data points with vector magnitude equal to 1.

Point 1: A= [1,2]T, slope A= 60 deg.

Point 2: B= [3,1]T, slope B = 30 deg.

Solution

Solution

Solution

Solution

1

Example problem On changing the tangent vector of the curve, the shape of the

curve changes accordingly. Using the data given in the last problem, a plot can be made given that the magnitude of the tangent vector equals 1,2,3,6 and 12 respectively.

Following the procedure of the earlier example we can find the Hermite curve family equations with different magnitude of tangent vectors. The plots are following:

Bezier Curves

The Hermite curve discussed in the previous section is based on interpolation techniques.

On the contrary, Bezier curves are based on approximation techniques that produce curves which do not pass through all the given data points except the first and the last control point.

A Bezier curve does not require first order derivative; the shape of the curve is controlled by control points.

Bezier Curves

As in the previous section, we consider here one segment of the curve.

For n+1 control points, the Bezier curve is defined by a polynomial of degree n as follows:

Bezier Curves

Here V0, V1, . Vn are position vectors of n+1 points (V0, V1, . Vn in Figure 3.19) that form the so-called characteristic polygon of the curve segment.

Properties of Bezier Curves

The curves pass through the first and last control points (V0 and Vn from the preceding function) at parameter values 0 and 1. In the figure on the left the starting point V1 of the second line and the endpoint V4 of the first line have the same position.

The tangents at the first and last points are in the directions of the first and last segments of the characteristic polygon. This can easily be seen:

V(0) = n *V(1)-V(0)+, V(1) = n *V(n)-V(n-1)]

Where [V(1) V(0)] and [V(n)-V(n-1)] define the first and last segments of the curve polygon. This implies that by aligning the last control point of the first Bezier curve segment, the connection point, and the first control point of the next curve segment will result in C1 continuity between the two curve segments.

Proof of the V(0) and V(1) values The Bezier curve has the convex hull property. By convex hull property we mean that the entire curve lies within the characteristic polygon. This property is useful when curve intersection and spatial bounds on the curve segments are calculated.

Example problem

Develop the equation of a Bezier curve, find the points on the curve for t = 0, , , , and 1, and plot the curve for the following data. The coordinates of the four control points are given by:

V0 = [0,0,0], V1= [0,2,0], V2= [4,2,0], V3= [4,0,0]

Solution

B-Spline, Rational B-Spline, and Non uniform rational B-Spline curves

The B-Spline is considered a generalization of the Bezier curve.

Local control is an interesting feature of B-Spline curves which implies that any change in the local control point affects only part of the curve.

Rational B-splines are generalizations of B-Splines. Interestingly, an RBS has an added parameter (called a weight) associated with each control point to control the behavior of the curve.

Surface Modeling

In wireframe modeling, we take advantage of the simplicity of certain surfaces. For example a plane is represented by its boundaries. We say nothing about the middle of the plane, which is fine because we known that the middle of a plane is till a plane.

It is common knowledge that the shapes of the cars, aircrafts and ships are very complex and do not consist of simple and regular geometric shapes. So, it is not easy to represent them by wireframe models and so surface modeling is prefered.

Surface modeling system contains definition of surfaces, edges and vertices. It contains all the information that a wireframe does and in addition it also contains the information of how the two surfaces connect to each other.

Surface entitiesPlane surface:

A plane surface is the simplest surface that is defined by three non coincident points or its variation.

Ruled Surface:

A ruled surface can be defined as the linear interpolation between two general curves. Informally the straight lines connecting the two rails form the surface.

Surface of revolution: This surface is generated by rotating a planar curve in space

about an axis at a certain angle.

Tabulated cylinder: This surface is generated by sweeping a planar curve in space in a

certain direction at a certain distance. In the example above a straight line sweeps about

a path that is circle and forms the surface. The straight line is called the generatrix and

the circle is the directrix.

Bezier surface and B-Spline surface: Bezier and B-spline surfaces are both synthetic

surfaces. Like synthetic curves a synthetic surface approximates the given input data.

The Bezier and B-Spline surfaces are also formed in the same manner as the 2D

curves.

Surface representations In case of curves we have seen the representation of curves by

implicit/explicit equations.

Implicit equation to describe a surface:

F(x, y, z) = 0, Its geometric meaning is that the locus of points that satisfy the constraint equation defines the surface.

Explicit equation to describe a surface:

V = [x,y,z]T = [x,y,f(x,y)] T where V is the position of a variable point on the surface. In this equation, we directly define the variable oint coordinates x,y,z. The z-coordinates of the position vector of the variable points are defined by x,y through a suitable function f(x,y) as shown in the figure below.

Comparing the equations for a 2-D curve and a 3-D surface the only difference between a space curve

and a surface mathematically is that points on a

space curve are defined by a single degree of

freedom by that on a surface have two degrees of

freedom.

Usually an arbitrary surface is defined in x,y with a functional relation f(x,y) by an x-y grid with P+1.Q+1

points.

Parametric Equation/ Representation of a Surface/ synthetic surface

There are no extra parameters in equations represented earlier and as such these are called non-parametric representation of equations. The corresponding equations that utilize parameters are called parametric equations and have two degrees of freedom and are represented as :

V(s,t) = [x,y,z]T = [X(s,t), Y(s,t), Z(s,t)]T , smin< s< smax , tmin< t< tmax

Where x,y and z are functions of two parameters s and t.

Hermite bicubic surface: Surfaces are normally defined in patches, each patch corresponds to a rectangular domain in s-t space, just as we discussed the s-domain in the previous section. Surface patches are dealt with in the same way; however, patches are much more complicated than segments.

Just as there is a characteristic third order equation to describe a two dimensional curve there is a third order 16 term series used to describe the cubic parametric equation for a surface.

Parametric representation of surface

Parametric representation of a surface