B-spline Curves

Q: Explain the B-spline curves. What are the properties of B-spline curves.

Ans:-

Spline  :-

• Drafting terminology

Spline is a flexible strip that is easily flexed to pass through a series of design points (control points) to produce a smooth curve. 

• Spline curve – a piecewise polynomial (cubic) curve whose first and second derivatives are continuous across the various curve sections.

Spline Representations

 • A spline is a smooth curve defined mathematically using a set of constraints 

• Splines have many uses:

 • 2D illustration 

• Fonts 

• 3D Modelling

 • Animation

Big Idea:-

• User specifies control points

• Defines a smooth curve

Properties of B-spline Curve

B-spline curves have the following properties −

• The sum of the B-spline basis functions for any parameter value is 1.
• Each basis function is positive or zero for all parameter values.
• Each basis function has precisely one maximum value, except for k=1.
• The maximum order of the curve is equal to the number of vertices of defining polygon.
• The degree of B-spline polynomial is independent on the number of vertices of defining polygon.
• B-spline allows the local control over the curve surface because each vertex affects the shape of a curve only over a range of parameter values where its associated basis function is nonzero.
• The curve exhibits the variation diminishing property.
• The curve generally follows the shape of defining polygon.
• Any affine transformation can be applied to the curve by applying it to the vertices of defining polygon.
• The curve line within the convex hull of its defining polygon.

Q: What are the parametric continuity conditions in a spline representation?

Ans:-

We imposed various continuity conditions at the connection points to ensure smooth transition from one section to the next section of the piece wise parametric curve. If the spline curve is described in terms of parametric coordinate form like

                            x = x(u), y = y(u), z = 2 (U)

Spline curve can be represented as:

1. Zero-order parametric continuity : When two pieces of the curve meet at a point, this is known as zero-order parametric continuity as shown in Fig. 1, and is represented by Cº.

 Fig..1. Zero-order,

The values of x, y, z can be evaluated at v, for the first piece of curve and will be equal to the values ofx,y,z evaluated at v, for the second curve.

2. First-order parametric continuity : The first-order parametric continuity shows that the first derivatives of the coordinate function x = x (u), y = y(u), z = 2 (v) for two successive pieces are equal at their joining point as shown in Fig.2. This is represented by C'.

Fig. 2. First-order.

3.Second order parametric continuity :-

The second order parametric continuity shows that both the first and second order derivation of the two curve pieces are the same at intersection as shown in Fig.3. This is represented by c2

Q: Explain the characteristics of B-spline curve. How it  is useful in interpolation?

Ans:-

Characteristics of B-spline curve :

1. The sum of the B-spline basis functions for any parameter value

      is 1.

2. Each basis function is positive or zero for all parameter values.

3. Each basis function has precisely one maximum value, except for

= 1.

4. The maximum order of the curve is equal to the number of vertices of defining polygon.

5. The degree of B-spline polynomial is independent on the number ofvertices of defining polygon.

6. The curve exhibits the variation diminishing property.

7. The curve generally follows the shape of defining polygon.

B-spline curve is useful for interpolation because :

1.  The degree of B-spline polynomial is independent of the number ofcontrol points of defining polygon.

2.  B-spline allows local control over the curve surface because each controlpoint affects the shape of a curve only over a range ofparametersvalues where its associated basis function is non-zero.

Q: Describe parametric representations of surface?

Ans:-

Parametric representation of surfaces means a continuous, vector-valuedfunction P(s,t) of two variables, or parameters s and t where the variablesare allowed to range over connected region of the uv plane.

Parametric description :

1. The three equations for determining the coordinates of any point on thesurface S are described in terms of parameters, say, s and f, and inparameter ranges [a, b] and [c, d], which may be infinite :

x = f(s, t),

S: y = g(s, t).

z = h(s, t)

2. The coordinates of any point P on the surface have the form {F(s,f) , g(s,t) , h(s,t)}

3. Spherical surface in parametric form can be defined in terms of the Angular parameter θ and  ϕ :

x=r cos ϕ cos θ, for – π/2 <= ϕ <= π/2

y = r cos ϕ sin θ, for - π<= θ <= π

z = r sin ϕ

4. Parametric representation for a torus is similar to those for an ellipse, except that the angle ϕ extends over 360°. Using latitude and longitude angles θ and ϕ, we can describe the torus surface as the set of points that satisfy

x=rx cos ϕ cos θ, for – π/2 <= ϕ <= π/2

y = ry cos ϕ sin θ, for - π<= θ <= π

z = rzsin ϕ

Q: Write note on interpolation.

Ans:- 

1. Interpolation is a method of constructing new data points within the range of a discrete set of known data points.

2. The number of data points obtained by sampling or experimentationrepresents the values of a function for a limited number of values ofthe independent variable.

3. It is often required to interpolate, i.e., estimate the value of that functionfor an intermediate value of the independent variable.

4. A few data points from the original function can be interpolated to produce a simpler function which is still fairly close to the original.The resulting gain in simplicity may outweigh the loss from interpolation error.

5. The main task in this process is to find the suitable mathematical expression for the known curve.

6. Interpolation technique is used when we have to draw the curve by determining intermediate points between the known sample points.

Q: Explain the type of interpolation method.

Ans:- 

Types of interpolation method are :-

1.IDW:

a The IDW (Inverse Distance Weighted) tool uses a method of interpolation that estimates cell values by averaging the values of sample data points in the neighbourhood of each processing cell.

b. The closer a point is to the centre of the cell being estimated the more influence or weight it has in the averaging process.

c. A specified number of points or all points within a specified radius can be used to determine the output value of each location.

2 Kriging :

a. Kriging is a geostatistical procedure that considers both the distance and the degree of variation between known data points when estimating values in unknown areas.

b. This procedure generates an estimated surface from a scattered set of points with z-values.

c. It should be done before we select the best estimation method forgenerating the output surface.

3. Natural neighbour :

a. Natural neighbour interpolation finds the closest subset of inputsamples to a query point and applies weights to them based onproportionate areas to interpolate a value.

b. It is also known as Sibson or "area-stealing” interpolation.

4 Spline: The spline tool uses an interpolation method that estimates values using a mathematical function that minimizes overall surface curvature, resulting in a smooth surface that passes exactly through the input points.

5. Spline with barriers : The spline with barriers tool uses a methodsimilar to the technique used in the Spline tool, with the major differencebeing that this tool honours discontinuitiesencoded in both the inputbarriers and the input point data.

6. Topo to raster: It use an interpolation technique specifically designed to create a surface that more closely represents a natural drainage surface and better preserves stream networks from input contour data.

7. Trend:

(a.) Trend is a global polynomial interpolation that fits a smooth surface defined by a mathematical function (a polynomial) to the input sample points.

(b.) The trend surface changes gradually and captures coarse-scale patterns in the data.