Boundary Representations 1:
Fundamentals of NURBS surface representations
Ir. Pirouz Nourian
PhD candidate & Instructor, chair of Design Informatics, since 2010
MSc in Architecture 2009
BSc in Control Engineering 2005
MSc Geomatics, GEO1004, Directed by Dr. Sisi Zlatanova
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[Geometric, Topologic] Spatial Data Models Representations
Different terminologies and jargons! Some common grounds
• Computer Graphics (mainly concerned with visualization)
• Computational Geometry (algorithmic geometry)
• AEC {CAD, CAM, BIM} (architectural engineering and construction)
• CAD=: Computer Aided Design
• CAM=: Computer Aided Manufacturing
• BIM=: Building Information Modeling
• GIS: how can we represent geometric objects in large scale properly and
consistently?
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Categories of 3D Geometry Representations
Interior included or only the closure ? A note on our terminology
• Volume Representation:
1. Tetrahedral Meshes
2. Voxel Models
• Boundary Representation:
[AKA Surface Representation]
1. Polygon Mesh Models (Simple Brep)
2. [complex] B-rep* Models (NURBS patches)
* B-rep here refers to a specific class of boundary representations composed of advanced faces (as implemented in Rhino):
• ISO 10303-514 Advanced boundary representation, a solid defining a volume with possible voids that is composed by
advanced faces
• ISO 10303-511 Topologically bounded surface, definition of an advanced face, that is a bounded surface where the surface is
of type elementary (plane, cylindrical, conical, spherical or toroidal), or a swept surface, or b spline surface. The boundaries
are defined by lines, conics, polylines, surface curves, or b spline curves
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Boundary Representation
Representing high dimensional objects with lower dimensional primitives
1. Polygon Mesh ≡ Simple B-rep=composed of straight/flat elements
We will discuss them later in depth
1. NURBS patches ≡[complex] B-rep*=composed of curved elements
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Why NURBS?
advantages and disadvantages for GIS?
• Known and used in AEC {CAD, BIM}
• In GIS ?
Bilbao Guggenheim Museum
Bus stop near Sebastiaansbrug Delft
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NURBS Representation
Non Uniform Rational Basis Splines (NURBS) are used
for modeling free-form geometries accurately
• An elegant mathematical description of a physical drafting aid as a (set of)
parametric equation(s).
Image courtesy of Wikimedia
Image courtesy of http://www.boatdesign.net
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Splines in Computer Graphics
All types of curves can be modeled as splines
•
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NURBS Representation
Non Uniform Rational Basis Splines (NURBS) are used
for modeling free-form geometries accurately
• offer one common mathematical form for both, standard analytical shapes (e.g. conics) and
free form shapes;
• provide the flexibility to design a large variety of shapes;
• can be evaluated reasonably fast by numerically stable and accurate algorithms;
• are invariant under affine as well as perspective transformations;
• are generalizations of non-rational B-splines and non-rational and rational Bezier curves and
surfaces.
From:
http://web.cs.wpi.edu/~matt/courses/cs563/talks/nurbs.html
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Parametric Curves in General
How do numeric weights correspond to physical
weights?
𝑥(𝑡)
𝑃 𝑡 = 𝑦(𝑡)
𝑧(𝑡)
𝑡
Example 1: 𝑌 = 𝑋 + 1 → 𝑃 𝑡 = 𝑡 + 1
0
𝐶𝑜𝑠(𝑡)
Example 1: 𝑋 2 + 𝑌 2 = 1 → 𝑃 𝑡 = 𝑆𝑖𝑛(𝑡)
0
𝐶𝑜𝑠 2 (𝑡) + 𝑆𝑖𝑛2 (𝑡) = 1
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NURBS equations
All from a summary by Markus Altmann:
http://web.cs.wpi.edu/~matt/courses/cs563/talks/nurbs.html
𝐶 𝑢 =
where
𝑛
𝑖=0 𝑊𝑖 . 𝑃𝑖 . 𝑁𝑖,𝑘 (𝑢)
𝑛
𝑖=0 𝑊𝑖 . 𝑁𝑖,𝑘 (𝑢)
𝑡ℎ𝑒 𝑐𝑢𝑟𝑣𝑒 𝑒𝑣𝑎𝑙𝑢𝑎𝑡𝑒𝑑 𝑎𝑡 𝑝𝑎𝑟𝑎𝑚𝑒𝑡𝑒𝑟 𝑢
𝑊𝑖 : weights
𝑃𝑖 : control points (vector)
𝑁𝑖,𝑘 : normalized B-spline basis functions of degree k
These B-splines are defined recursively as:
𝑢 − 𝑡𝑖
𝑡𝑖+𝑘+1 − 𝑢
𝑁𝑖,𝑘 𝑢 =
× 𝑁𝑖,𝑘−1 𝑢 +
× 𝑁𝑖+1,𝑘−1 𝑢
𝑡𝑖+𝑘 − 𝑡𝑖
𝑡𝑖+𝑘+1 − 𝑡𝑖+1
and
1, 𝑖𝑓 𝑡𝑖 ≤ 𝑢 < 𝑡𝑖+1
𝑁𝑖,0 𝑢 =
0,
𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒
Where 𝑡𝑖 are the knots forming a knot vector
𝑈 = {𝑡0 , 𝑡1 , … , 𝑡𝑚 }
Note: 𝑘 enumerates 0 to degree, 𝑚 = 𝑛 + 𝑘 + 1
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NURBS interpolation
All contenet from Raja Issa
[Essential Mathematics for Computational Design]
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A weighted NURBS curve
How do numeric weights correspond to physical
weights?
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NURBS Objects
Non Uniform Rational Basis Splines are used for
accurately modeling free-form geometries
• 1D: Curves (t parameter)
• 2D: Surfaces (u & v parameters)
• 3D: B-Reps (each face is a surface)
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Parametric Space
Images courtesy of David Rutten, from Rhinoscript 101
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Parametric Locations:
1D Objects (Curves): u parameter (AKA as t parameter)
• Point at that address (𝐶(𝑢))
• Tangent vector
• Derivatives (𝐶 ′ (𝑢), 𝐶 ′′ (𝑢))
• Curvature
2D Objects (Surfaces): u,v parameters
• Point at that address (𝑆(𝑢, 𝑣))
• Normal vector (𝑁(𝑢, 𝑣))
• Curvature
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1D Curvature: Vector or Scalar?
•
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Continuity
• G0 (Position continuous)
• G1 (Tangent continuous)
• G2 ( Curvature Continuous)
Images courtesy of Raja Issa, Essential Mathematics for Computational Design
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(1D Curvature Analysis)
• Discretization
Segments
• Measurement
at the middle of each
segment
• Attribution
to each segment
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Surface Curvature
•
Images courtesy of Raja Issa, Essential Mathematics for Computational Design
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Surface Continuity: Zebra Analysis
Images courtesy of Raja Issa, Essential Mathematics for Computational Design
• Open question: How can we measure curvature on meshes?
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(2D Curvature Analysis: NURBS surface)
• Discretization
Sub-surfaces
• Measurement
At UV points
• Attribution
To sub-surfaces
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We will now see a NURBS data model…
Questions? p.nourian@tudelft.nl
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