6: Bézier Curves and Surface Continuity

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Questions and Answers

What does C1 continuity between two curves ensure?

The curves share a common endpoint and have the same slope. (correct)

The curves are joined end to end without any shared points.

The curves share a common endpoint and have the same curvature.

The curves are tangent to a straight-line network at their endpoints.

How is the degree of a Bézier curve determined?

By the area enclosed by control points.

By the length of the curve.

By the number of control points. (correct)

By the curvature at its endpoints.

What does C2 continuity require between two curves?

The curves have different endpoints and different slopes.

The curves share a common endpoint, the same slope, and the same curvature. (correct)

The curves have the same endpoint but different curvatures.

The curves have the same slope but different endpoints.

What is true about the end points of a Bézier curve?

They are the points through which the curve passes exactly. Signup and view all the answers

What is one characteristic of Bézier surfaces compared to Bézier curves?

They pass through four corner points instead of two endpoints. Signup and view all the answers

What is the primary goal of using B-spline curves in engineering design?

To achieve better local control Signup and view all the answers

What characteristic differentiates B-spline curves from Bézier curves?

B-spline curves consist of polynomial segments Signup and view all the answers

What does the symbol $n$ represent in the B-spline formula?

The number of control points Signup and view all the answers

How is the B-spline curve $p(t)$ mathematically expressed?

Through the weighted average of control points using basis functions Signup and view all the answers

What does the knot vector $u$ define in the context of B-spline curves?

The intervals where the polynomial segments join Signup and view all the answers

In a B-spline, the basis functions $N_{i,k}(t)$ are dependent on which variable?

The value of $t$ in the interval $u$ Signup and view all the answers

What degree do B-spline curves typically have in relation to control points?

One less than the number of control points Signup and view all the answers

What key advantage do B-spline curves offer in computer-aided design?

They provide local control without affecting the entire curve Signup and view all the answers

What represents the degree of a Bézier curve?

Number of control points minus one Signup and view all the answers

What are Bézier curves primarily used for?

To approximate real continuous functions Signup and view all the answers

Which polynomial functions are used in defining Bézier curves?

Bernstein polynomials Signup and view all the answers

What does the formula 𝐩 𝑡 = 𝑛∑𝑖=0 𝐵𝑖,𝑛 𝑡 𝐏𝑖 represent?

A point on a Bézier curve Signup and view all the answers

What does the variable 't' represent in the context of Bézier curves?

A continuous interval value between 0 and 1 Signup and view all the answers

In the equation 𝑝(𝑡) = 1 − 𝑡²𝐏0 + 2𝑡(1 − 𝑡)𝐏1 + 𝑡²𝐏2, what does '2𝑡(1−𝑡)' represent?

The basis function for a quadratic Bézier curve Signup and view all the answers

How is a Bézier curve defined mathematically?

Using a finite sum of control points Signup and view all the answers

What is the behavior of the Bernstein polynomials for Bézier curves?

They vary between 0 and 1 depending on 't' Signup and view all the answers

What is a key characteristic of Bézier curves compared to traditional curves?

They are formed by using control points Signup and view all the answers

What is the degree of the B-spline curve segment 𝐩4 (𝑡) if 𝑘 = 3?

3 Signup and view all the answers

What type of knot vector results when the knot intervals are of equal lengths?

Uniform Signup and view all the answers

What does increasing the number of control points affect in a B-spline curve?

The overall shape of the curve Signup and view all the answers

If the number of control points is set equal to 𝑘, what kind of curve is represented?

Bézier curve Signup and view all the answers

How does a control point affect a B-spline curve?

It only affects local areas defined by the curve degree Signup and view all the answers

What is the continuity at a knot in terms of degree 𝑘?

C𝑘−2 Signup and view all the answers

How many separate curves can be affected by a single control point 𝐏𝑖?

Up to 𝑘 Signup and view all the answers

What is the effect of moving a control point on the associated curve?

It only affects the segment connected to it Signup and view all the answers

What is the effect of setting the value of $k$ in B-spline curves?

It determines the degree of the curve segments. Signup and view all the answers

In the context of B-spline curves, what does the term 'control polygon' refer to?

A geometric figure formed by joining the control points. Signup and view all the answers

What is the significance of having multiple knots in B-spline curves?

It allows for smoother transitions between curve segments. Signup and view all the answers

For a B-spline with $5$ control points and $k = 3$, what is the resulting value of $n+k$?

$7$ Signup and view all the answers

What is the term used for the parameter that defines intervals in B-spline curves?

Knots Signup and view all the answers

What geometric condition is fulfilled when the control polygon coincides with curve segments?

The segments are straight lines. Signup and view all the answers

When manipulating B-spline curves, what aspect does changing the knot vector influence?

The complexity and continuity of the curve. Signup and view all the answers

In end-point interpolation for B-spline curves, which statement is true regarding control points?

Control points determine the curve's tangent at endpoints. Signup and view all the answers

What is the primary characteristic of a B-spline curve with $k = 2$?

B-spline curves of degree one resulting in straight lines. Signup and view all the answers

What does the notation $t ∈ u_{k-1}, u_{n+1}$ indicate in the context of B-spline curves?

It shows the range of parameter values for the curve. Signup and view all the answers

Study Notes

Continuity Between Curves and Surface Patches

Curves and surface patches can be combined to model a composite curve or surface by enforcing constraints on continuity.
C0 (0th order) - Two curves are joined end to end and share a common end point.
C1 (1st order) - Two curves are joined end to end and share a common end point as well as the same slope at this point.
C2 (2nd order) - Two curves are joined end to end and share a common end point as well as the same slope and curvature at this point.

Bézier Curves & Surfaces

Bézier curves are defined by a network of control points and are either in approximated or interpolated form.
The degree of the curve is determined by the number of control points.
The curves pass exactly through the two endpoints (curve) or four corner points (surface).
These curves are tangent to the straight-line network at the end points (curve) or corner points (surface).

Bézier Curves: Definition and Basis Function

Bézier curves use polynomial functions to approximate real continuous functions on a closed interval.
A point p on a Bézier curve is defined by a sum of basis functions (Bernstein polynomials) multiplied by their corresponding control points.
The degree of a Bézier curve is determined by the number of control points minus 1.

Splines

Splines are often employed in CAD for creating smooth curves.
They use composite polynomial functions and are commonly implemented using B-splines.

B-spline Curves

B-splines are a generalized form of Bézier curves.
They consist of polynomial segments of degree k-1 with a network of control points.
B-splines aim to achieve better local control.
Each curve segment is controlled by a specific number of control points, depending on the degree k.
The control polygon coincides with the curve segments.

B-spline Curves: Examples

For a degree of 1 (k-1=1) with 5 control points, each curve segment is a straight line controlled by 2 control points.
For a degree of 2 (k-1=2) with 5 control points, each curve segment is a quadratic curve controlled by 3 control points.
For a degree of 3 (k-1=3) with 5 control points, each curve segment is a cubic curve controlled by 4 control points.

B-spline Curves: End-point Interpolation

End-point interpolation in B-splines ensures the curve passes through the first and last control points.
This is achieved by assigning a multiplicity of k to the starting and ending knots in the knot vector.

Splines in NX

NX software offers various spline types with varying polynomial degrees, including linear, quadratic, cubic, and 4th degree splines.

B-spline Curves & Surfaces II

Local control dictates that moving one control point only affects a localized section of the curve or surface patch depending on the degree k.
Approximated and interpolated B-splines exist.

B-spline Curves: Summary

B-splines are composed of separate curve segments joined at the points defined by knot values.
The knot vector is uniform if the knot intervals are of equal length.
The degree of each curve segment is k-1.
Increasing the number of control points does not change the degree of each curve segment.
The continuity (C) at the knot is equal to k-2.
Each curve segment is influenced by k control points.
A control point can affect a maximum of k curve segments.
If the number of control points equals k, the B-spline curve becomes a Bézier curve.

B-spline Curves: Control Point Influence

For a B-spline curve with a degree of 3 (k=3), the curve segment p4(t) is influenced by control points P1, P2, P3, and P4.

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Description

This quiz explores the concepts of continuity between curves and surface patches, including C0, C1, and C2 continuity. Additionally, it covers the fundamentals of Bézier curves and surfaces, focusing on their definition, control points, and basis functions. Test your understanding of these essential topics in computer graphics and geometrical modeling.