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?

<p>They are the points through which the curve passes exactly. (D)</p> Signup and view all the answers

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

<p>They pass through four corner points instead of two endpoints. (A)</p> Signup and view all the answers

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

<p>To achieve better local control (B)</p> Signup and view all the answers

What characteristic differentiates B-spline curves from Bézier curves?

<p>B-spline curves consist of polynomial segments (B)</p> Signup and view all the answers

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

<p>The number of control points (D)</p> Signup and view all the answers

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

<p>Through the weighted average of control points using basis functions (D)</p> Signup and view all the answers

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

<p>The intervals where the polynomial segments join (C)</p> Signup and view all the answers

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

<p>The value of $t$ in the interval $u$ (D)</p> Signup and view all the answers

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

<p>One less than the number of control points (B)</p> Signup and view all the answers

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

<p>They provide local control without affecting the entire curve (B)</p> Signup and view all the answers

What represents the degree of a Bézier curve?

<p>Number of control points minus one (D)</p> Signup and view all the answers

What are Bézier curves primarily used for?

<p>To approximate real continuous functions (C)</p> Signup and view all the answers

Which polynomial functions are used in defining Bézier curves?

<p>Bernstein polynomials (B)</p> Signup and view all the answers

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

<p>A point on a Bézier curve (C)</p> Signup and view all the answers

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

<p>A continuous interval value between 0 and 1 (D)</p> Signup and view all the answers

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

<p>The basis function for a quadratic Bézier curve (C)</p> Signup and view all the answers

How is a Bézier curve defined mathematically?

<p>Using a finite sum of control points (D)</p> Signup and view all the answers

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

<p>They vary between 0 and 1 depending on 't' (B)</p> Signup and view all the answers

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

<p>They are formed by using control points (A)</p> Signup and view all the answers

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

<p>3 (C)</p> Signup and view all the answers

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

<p>Uniform (D)</p> Signup and view all the answers

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

<p>The overall shape of the curve (B)</p> Signup and view all the answers

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

<p>Bézier curve (C)</p> Signup and view all the answers

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

<p>It only affects local areas defined by the curve degree (B)</p> Signup and view all the answers

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

<p>C𝑘−2 (D)</p> Signup and view all the answers

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

<p>Up to 𝑘 (B)</p> Signup and view all the answers

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

<p>It only affects the segment connected to it (D)</p> Signup and view all the answers

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

<p>It determines the degree of the curve segments. (D)</p> Signup and view all the answers

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

<p>A geometric figure formed by joining the control points. (A)</p> Signup and view all the answers

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

<p>It allows for smoother transitions between curve segments. (D)</p> Signup and view all the answers

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

<p>$7$ (B)</p> Signup and view all the answers

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

<p>Knots (C)</p> Signup and view all the answers

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

<p>The segments are straight lines. (B)</p> Signup and view all the answers

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

<p>The complexity and continuity of the curve. (A)</p> Signup and view all the answers

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

<p>Control points determine the curve's tangent at endpoints. (D)</p> Signup and view all the answers

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

<p>B-spline curves of degree one resulting in straight lines. (C)</p> 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?

<p>It shows the range of parameter values for the curve. (D)</p> Signup and view all the answers

Which of the following represents a type of 3D model?

<p>3D solid models (D)</p> Signup and view all the answers

2D models cannot represent volume.

<p>True (A)</p> Signup and view all the answers

What are Bézier surfaces primarily composed of?

<p>Bézier patches</p> Signup and view all the answers

A wireframe model consists of points and _____ to represent 3D structures.

<p>edges</p> Signup and view all the answers

Match the following types of models with their characteristics:

<p>3D wireframe model = Uses lines to connect points 3D surface model = Represents surfaces without volume 3D solid model = Includes faces and volumes 2D drawings = Flat representations of objects</p> Signup and view all the answers

Which of the following is NOT a numerical form used to represent surfaces?

<p>Quadratic equations (B)</p> Signup and view all the answers

Surface models have no information about ______.

<p>volume</p> Signup and view all the answers

Name one advantage of using B-spline curves in engineering design.

<p>Flexibility and control in shape design.</p> Signup and view all the answers

Which statement about a Bézier curve defined by 100 control points is correct?

<p>The curve has a degree of 99. (D)</p> Signup and view all the answers

Moving a single control point affects only a small portion of the Bézier curve.

<p>False (B)</p> Signup and view all the answers

What is the relationship between the number of control points and the degree of a Bézier curve?

<p>The degree of a Bézier curve is the number of control points minus one.</p> Signup and view all the answers

The degree of a Bézier curve depends on the number of control points and is calculated as the number of control points minus ___.

<p>1</p> Signup and view all the answers

Match the following statements with their correct characteristics regarding Bézier curves:

<p>A. Degree of the curve = 1. Number of control points minus one B. Impact of moving a point = 2. Affects the entire curve C. Calculation effort = 3. Increases with more control points</p> Signup and view all the answers

Which of the following is true about Bézier curves?

<p>Moving a control point will necessitate recalculating the entire curve. (B)</p> Signup and view all the answers

The calculation effort to compute a Bézier curve remains constant despite the number of control points.

<p>False (B)</p> Signup and view all the answers

How does the complexity of a Bézier curve change with the addition of control points?

<p>The complexity increases with the number of control points.</p> Signup and view all the answers

What is the degree of a Bézier curve if it has 5 control points?

<p>4 (C)</p> Signup and view all the answers

A Bézier curve can only be defined using linear polynomials.

<p>False (B)</p> Signup and view all the answers

What is the formula that represents a Bézier curve based on its control points?

<p>p(t) = Σ B_{i,n}(t) P_i</p> Signup and view all the answers

The basis functions used in Bézier curves are known as __________ functions.

<p>Bernstein</p> Signup and view all the answers

Match the following components related to Bézier curves with their descriptions:

<p>Control Points = Points that determine the shape of the curve Degree = Number of control points minus one Bézier Curve = A continuous function defined by control points Basis Functions = Mathematical functions used for weights in the curve</p> Signup and view all the answers

What characteristic does the sum of the basis functions add up to for any value of 't'?

<p>1 (A)</p> Signup and view all the answers

A Bézier curve can have a degree greater than the number of its control points.

<p>False (B)</p> Signup and view all the answers

How does the control point number affect the shape of a Bézier curve?

<p>Increasing the number of control points allows for more complex shapes.</p> Signup and view all the answers

In the equation 𝑝(𝑡) = 1 − 𝑡²𝐏0 + 2𝑡(1 − 𝑡)𝐏1 + 𝑡²𝐏2, '𝑡' represents __________.

<p>the parameter that varies between 0 and 1</p> Signup and view all the answers

What is the purpose of using Bézier curves in computer graphics?

<p>To create smooth curves and shapes (C)</p> Signup and view all the answers

What continuity condition is set for the first section curve?

<p>G1 (D)</p> Signup and view all the answers

The last section of a spline achieves G0 continuity.

<p>False (B)</p> Signup and view all the answers

What type of operation is reflected based on the splines?

<p>extrude operation</p> Signup and view all the answers

The middle section curve is set to ______ continuity.

<p>G0</p> Signup and view all the answers

Match the spline continuity condition to its description:

<p>G0 = Position continuity G1 = Tangent continuity G2 = Curvature continuity</p> Signup and view all the answers

Which section curve achieves curvature continuity?

<p>Last Section (C)</p> Signup and view all the answers

In creating surfaces, the reflections show that the continuity is G2.

<p>False (B)</p> Signup and view all the answers

Creating splines in NX involves creating section curves that include lines and ______.

<p>splines</p> Signup and view all the answers

Which category of curves offers the most possibilities to change the shape of the curve?

<p>NURBS (B)</p> Signup and view all the answers

Freeform surfaces can often be created using extrude, revolve, or sweep operations.

<p>False (B)</p> Signup and view all the answers

What continuity condition is described as 'G2'?

<p>Curvature continuity</p> Signup and view all the answers

Freeform surfaces are widely used for bodywork, wings, and __________.

<p>forged parts</p> Signup and view all the answers

Match the following terms with their correct descriptions:

<p>Bézier Curves = Simple control point based curves B-Splines = Flexibly defined curves with multiple segments NURBS = Advanced curves allowing rational representations</p> Signup and view all the answers

What is the primary purpose of using reflections in CAD while modeling surfaces?

<p>To visualize surfaces and their continuity conditions (B)</p> Signup and view all the answers

Curves and surfaces can only be parametrized with Bézier curves.

<p>False (B)</p> Signup and view all the answers

Name one application of freeform surfaces.

<p>Car bodywork</p> Signup and view all the answers

What is the degree of a B-spline curve with control points equal to 5 and $k = 5$?

<p>4 (C)</p> Signup and view all the answers

In B-splines, increasing the number of control points results in a curve that has more flexibility and can better approximate complex shapes.

<p>True (A)</p> Signup and view all the answers

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

<p>knot vector</p> Signup and view all the answers

The degree of a Bézier curve is determined by the number of control points minus ___ .

<p>1</p> Signup and view all the answers

Match the following terms related to B-splines to their definitions:

<p>Control points = Points that define the shape of the curve Knot vector = Defines the intervals and segmentation of the curve Degree = One less than the number of control points Basis functions = Functions that determine the influence of control points on the curve</p> Signup and view all the answers

Which of the following is true regarding the continuity at a knot in a B-spline?

<p>It is always C0 continuous. (D)</p> Signup and view all the answers

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

<p>The interval of the parameter t for the B-spline curve.</p> Signup and view all the answers

A B-spline curve with $k = 2$ is always linear.

<p>True (A)</p> Signup and view all the answers

Flashcards

C0 Continuity

Two curves share a common endpoint.

C1 Continuity

Curves share endpoint and slope at that point.

C2 Continuity

Curves share endpoint, slope, and curvature.

Bézier Curve

Curve defined by control points, approximating/interpolating a function.

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Bézier Surface

Surface defined by control points, approximating/interpolating a function.

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Control Points (Bézier)

Points that shape the curve/surface.

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Degree (Bézier)

Control point count minus 1 for curves.

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Bernstein Polynomials

Basis functions used in Bézier definition.

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Spline

Smooth curve created with polynomial segments.

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B-spline

Generalized Bézier with better local control.

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Knot Vector

Values defining curve segment boundaries.

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Knot (B-spline)

Values in a B-spline curve's knot vector.

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Local Control

Changing one control point affects only part of the curve/surface.

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Degree (B-spline)

Determines the polynomial type of each segment.

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End-point Interpolation

Curve passes through first/last control points.

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Uniform Knot Vector

Knot intervals are equal.

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Continuity (C) at knot

Smoothness at a point where curve segments meet.

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Control Polygon

Line connecting control points in a B-spline.

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Approximating B-spline

Curve doesn't necessarily pass through control points.

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Interpolating B-spline

Curve passes through all control points.

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Control point influence (B-spline)

Number of control points influencing a segment.

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Bézier curve from B-spline

Occurs when the number of control points equals the degree.

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polynomial segments (B-spline)

Individual segments of different degrees combining to create the curves.

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Geometric Models

Representations of objects in 3D or 2D space, used for visualization and analysis.

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3D models

Models that represent objects in three dimensions, allowing for volume, shape, and surface details.

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2D models

Models that represent objects as flat shapes, providing information on their outline and dimensions.

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Wireframe Model

A basic 3D model consisting of only lines and points, representing edges and vertices.

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Surface Model

A 3D model that focuses on object surfaces, using curves and patches to represent shapes.

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Solid Model

A 3D model representing a solid object with volume, allowing for analysis of its mass and internal structure.

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Bézier Curves/Surfaces

Smooth curves/surfaces defined by control points with a mathematical formula, used for creating shapes in computer graphics.

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Solid Primitives

Basic building blocks for solid models like cubes, spheres, and cylinders, used to create complex shapes.

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Bézier Curve Definition

A Bézier curve is a mathematical curve that is defined by a set of control points. It approximates a real continuous function on a closed interval using polynomial functions.

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Basis Functions

Basis functions, also known as Bernstein polynomials, are the polynomial functions used to calculate the position of a point on a Bézier curve. These functions are weighted by control points.

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Control Points

Control points are a set of points used to define the shape of a Bézier curve. Each control point has a specific influence on the curve's path.

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Degree of a Bézier Curve

The degree of a Bézier curve is determined by the number of control points - 1. A higher-degree curve has more control points and can create a more complex shape.

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Bézier Curve Equation

The equation for a Bézier curve uses a weighted sum of control points multiplied by their corresponding basis functions. These basis functions are Bernstein polynomials.

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t ∈ [0, 1]

The parameter 't' in the Bézier curve equation ranges from 0 to 1. It represents the position along the curve, with 0 being the start point and 1 being the end point.

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Approximation

Bézier curves approximate a function on an interval. This means the curve gets close to the true function but doesn't exactly match it.

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Polynomial Functions

Bézier curves are constructed using polynomial functions. These functions ensure that the curve is smooth and continuous.

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Closed Interval

A Bézier curve is defined on a closed interval. This means the curve has a definite start and end point, and the parameter 't' is restricted to the interval.

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Continuous Function

A Bézier curve approximates a continuous function. This means the function has no breaks or jumps in its graph.

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B-spline Curve

A generalization of Bézier curves that uses B-splines to create smooth, flexible curves.

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B-spline Degree

The degree of the polynomial used for each segment of the B-spline curve. It determines the smoothness and flexibility of the curve.

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B-spline vs. Bézier Curve

Bézier curves are a special case of B-splines where the number of control points equals the degree. B-splines offer more flexibility and control.

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Control Points (B-spline)

Points that influence the shape of the B-spline curve, but the curve doesn't necessarily pass through all of them.

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Bézier Curve Degree

The degree of a Bézier curve is determined by subtracting 1 from the number of control points used to define it.

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Bézier Curve Calculation Effort

Calculating a Bézier curve requires more effort as the number of control points increases.

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Bézier Curve Global Control

Moving a single control point on a Bézier curve affects the entire curve's shape.

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Bézier Curve Control Point Influence

Each control point contributes to the overall shape of the Bézier curve.

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What is a Bézier curve?

A Bézier curve is a mathematical curve defined by a set of control points. The curve smoothly interpolates between these points, creating a smooth and predictable shape.

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

Moving a control point on a Bézier curve will change the shape of the entire curve. The closer a control point is to a particular section of the curve, the greater its influence on that section.

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How is the degree of a Bézier curve determined?

The degree of a Bézier curve is determined by the number of control points used to define it. The degree is always one less than the number of control points.

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What factors influence the complexity of calculating a Bézier curve?

The complexity of calculating a Bézier curve is influenced by the number of control points used to define it. More control points require more complex calculations, which can be computationally expensive.

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Spline Continuity

The smoothness of a curve at the point where different segments meet. It describes how well segments connect and how gradual the transitions are.

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Surface (Studio Surface)

A 3D shape created in a CAD program by connecting multiple curves together. It is a representation of the outer surface of an object.

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Section Curves

The individual curves that define a 3D surface. They are like the building blocks of the surface, representing the different edges or lines on the surface.

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NX

A CAD software package for creating 3D models, It is used by engineers and designers to build and analyze products.

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Reflections (in CAD)

A way to visualize how a surface is created. Reflections show how smooth or bumpy a surface is at different parts based on the section curves.

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Surface Continuity

How smoothly curves and surfaces connect. Different levels like C0 (shared endpoint), C1 (shared endpoint and slope), C2 (shared endpoint, slope, and curvature).

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Freeform Surfaces

Complex, curved surfaces in 3D modeling, not created by simple operations like extrude, revolve, or sweep.

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NURBS

Non-Uniform Rational B-Splines, a powerful modeling method combining B-Splines with rational functions for greater control.

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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.

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