CURVES DM2
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Lecture 2
Parametric - Most current CAD/CAM/CAE software utilizes a design feature called parametrics, a method of linking dimensions and variables to geometry in such a way that when the values change, the part changes as well. A parameter is a variable to which other variables are related, and these other variables can be obtained by means of parametric equations. In this manner, design modifications and creation of a family of parts can be performed in remarkably quick time compared with the redrawing required by traditional CAD. Parametric modification can be accomplished with a spreadsheet, script, or by manually changing dimension text in the digital model.
Questions ? This lecture will be asking questions about curves, their relationships to surfaces, and how they are used and controlled. Topics of discussion will be: Free form Curves Bezier Curves B-Spline Curves NURBS Curves Sub Division Curves Intersecting Curves.
CURVE SURFACE RELATION
CURVE SURFACE RELATION
| Curves are renders of mathematical equations. Curves in Rhino for instance are called NURBS Curves are renders of mathematical curves or Non - Uniform Rational B-Splines. (Non-Uniform Rational B-Splines). | Curves contain a Degree
equations.
| Curves may be used as profiles to generate surfaces. In most cases, the shape of the profile Curves in Rhino for instance are called curve heavily influences the final shape of the emerging surface.
NURBS curves
Curves contain: a degree
X
x
| They have: Control Points
They have: control points
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Curves may be used as profiles to generate surfaces. In most cases, the shape of the profile curve hea
CURVES CURVES
We can think of a curve as a connected one-dimensional series o editing and control.
These curves are called planar curves, in contrast to spatial curve | We can think of a curve as a connected one-dimensional series of points. That contain many options for editing and control. These Curves are called planar curves, in contrast to spatial curves such as helixes. If we use any 3rd degree NURBS curve and modify any set of points, the first curve segment to coincide again will be the one between the 2nd to last and the 3rd to last modified point. After that, all will coincide again.
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If we use any 3rd degree NURBS curve and modify any set of poin again will be the one between the 2nd to last and the 3rd to last
BEZIER CURVES | Before explaining NURBS, we will examine Bezier curves, Because NURBS is a generalization of a Bezier Curve | Bezier curves are parametric curves which are customizable and smooth. | Bezier curves are widely used in computer graphics to model smooth curves. As the curve is completely contained in the convex hull of its control points, the points can be graphically displayed and used to manipulate the curve intuitively. | The following figure shows a simple Bezier curve sequence (C), its control points (1), (2), (3), (4), and its control polygon (p). The control points are also called control handles. | Each point on a Bezier curve (and on many other types of curves) is computed as a weighted sum of all control points. This means that each point is influenced by every control point. The first control point has maximum impact on the beginning of the curve, the second one reaches its maximum in the first half of the curve, etc.
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BEZIER CURVES | Limitations
-Remember, Bezier curves are determined by their control points
-A Bezier curve with a large number of control points becomes impractical for design. The degree of the curve increases and the curve shape resembles less and less the shape of the control polygon.
| What are Bezier curves used for? These curves are mainly used in interpolation, approximation, curve fitting, and object representation. Beziers are also used as the principle drawing elements in softwares like illustrator, Photoshop, Flash, and Macromedia. An Example would be in Typeface design.
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B-SPLINE CURVES | They are free form curves that consist of Bezier Curve segments of the same degree and that are knotted together at their endpoints with the highest possible smoothness. It is important to note the two intuitive design parameter of the B-Spline: The Control Points The Curve Degree.
| When we speak of local control points, we mean that the control point within that zone will alter that part of the curve but not the whole thing. Remember that in a Bezier curve, movement of the curve all depends on the control points. In a B-spline curve, it all depends on the CURVE SEGMENTS and their respective zones.
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NURBS NURBS CURVES
CURVES
NON UNIFORM RATIONAL B-SPLINES
Non-Uniform B-Splines, are mathematical representations of 3-D geometry accuratelythat | NURBS OR Non Uniform Rational Rational B Splines are mathematical representations ofthat 3Dcan geometry describe any shape a simple line, circle, or curve to thearc, most complex 3-Dto organic free- comcan accurately describe any from shape from 2-D a simple 2D arc, line, circle, or curve the most plex 3D organic free form surface of solid. form surface or solid. What is a NURBS curve?
What is a NURBS curve?
A NURBS curve isby defined four things: degree,control control points, knots,knots and anand evaluation rule. A NURBS curve is defined fourbythings: degree, points, an evaluation rule. Degeee
Degree
The degree is a positive whole number The degree is a positive whole number. This number is usually 1,2,3, or 5, but can be any number is usually 1, 2, 3 or 5, but can be any positive whole number. positive whole This number.
NURBS lines and polylines are usually degree 1, NURBS circles are degree 2, and most free-form curves are degree 3 or 5. NURBS lines and polylines are usually degree 1, NURBS circles are dgree 2 and most free form curves are degree 3 or 5 Control Points One of easiest ways to change the shape of a NURBS curve is to move its Control Points control points. The control points have an associated number called a weight . With a few exceptions, weightsways are positive numbers.the Whenshape a curve’sof control points all have the One of the easiest to change a NURBS curve issame weight (usually 1), the curve is called non-rational, otherwise the curve is called to move its control points. rational. The control points have an associated number called a weight. With a few exceptions, weights are positive numbers. When a curves control points all have the same weight (usually 1), the curve is called non rational, otherwise the curve is called rational.
What are the parts of a curve?
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NURBS CURVES
URBS CURVES
NON UNIFORM RATIONAL B-SPLINES | Evaluation Rule
tion Rule
The NURBS evaluation rule is a formula that involves the degree, control points, and knots. you can think of the evaluation rule as a black box that eats a parameter and produces a point location.
RBS evaluation rule is a formula that involves the degree, control points, and knots.
The degree, knots, and control points determine how the black box works.
n think of the evaluation rule as a black box that eats a parameter and produces a point location.
a central spatial B-Spline Curve gree, knots, andNURBS control is points determineprojection how the blackof boxaworks.
ORM RATIONAL B-SPLINES
B-Spline Curves
mathematical representations of 3-D geometry that can accurately ine, circle, arc, or curve toWherin the most all complex 3-D organic weights arefreeequal
BS is a central projection of a spatial B-splin curve
Equal weights mean that the control points of the Associated spatial B-spline curve all have a constant Z-coordinate and are lying in the same horizontal plane
B-spline curves enjoy line curves are special NURBS s: degree, control points, knots, and an evaluation rule. (because they consist
the property of affine in variance of Bezier curve segments)
Wherein all weights are equal
tive whole number.
Equal weights mean that the control points of the Associated spatial B-spline curve all have a constant Z-coordinate and are lying in the same horizontal plane.
S circles are degree 2,
urve is to movecurves its B-spline
enjoy the property of affine in variance (because they consist of Bezier curve segments)
weight . With a few exceptions, points all have the same herwise the curve is called
What are the parts of a curve?
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NURBSNURBS CURVESCURVES
NON UNIFORM RATIONAL B-SPLINES
What are NURBS goodwhen for when designing? What are NURBS good for designing? To move To valuable geometric models between move valuable geometric models betweenvarions various modeling, modeling, rendering, animation, and engineering rendering, animation, and engineering analysis programs. programs. NURBS can accurately represent both standard geometric objects like lines, circles, ellipses, spheres, tori, and free-form geometry like car bodies and human bodies NURBS have a great deal of flexibility in manufacturing, and specifically digital fabrication.
NURBS can accurately represent both standard geometric objects like lines, circles, ellipses, spheres, and tori, and free-form geometry like car bodies and human bodies.
NURBS have a great deal of flexibility in manufacturing, and specifically digital fabrication.
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SJET Skylar Tibbits, in collaboration with Point b design, presented the Path Responsive Cabinet and Path Responsive Wall, Aug. 2008. The tool path is extracted from isocurves and undulates in response to surface depth, creating interlocking patterns on top of a given 3D surface. This utility was created to control the tool as it describes a cut path across any surface. This attempts to extend CAM software capabilities by specifying unique tool patterning through controlled step-over and bit depths, while still maintaining the desired 3D surface. Custom milled cabinets installed in a residence/personal gallery space outside Philadelphia, PA.
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The Spline Chair designer Unto
Ben Pimlott Bldg architect: WillAlsop
DownloadGridshell Architect: EdwardCullinan
the Hangar 7
architect: VolkmarBurgstaller