conference of phd students in computer science employing pythagorean hodograph curves for artistic...
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Conference of PhD Students in Computer Science
Employing Pythagorean Hodograph Curves for Artistic Patterns
Gergely Klár, Gábor Valasek
Eötvös Loránd UniversityFaculty of Informatics
June 29 - July 2, 2010Szeged, Hungary
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Goal
Create a tool to aid the design of aesthetical, fair curves
In particular design element creator for vines, swirls, swooshes and floral components
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Previous work
Floral components are popular elements in both ornamental and contemporary abstract design
Tools can aid among other things:– Generation of ornamental elements
– Generation of ornamental patterns
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Previous work
Plants generated with L-systems, proposed by Prusinkiewicz and Lindenmayer
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Previous work
Wong et. al. proposed a method for filling a region of interest with ornaments using proxy objects that can be replaced by arbitrary ornamental elements
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Previous work
Xu and Mould created ornamental patters by simulating a charged particle's movement in a magnetic field (magnetic curves)
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Our focus
We concentrate on using polynomial curves for element design
Let us presume that a pleasing curve has a smooth and monotone curvature Farin's definition:
A curve is fair if its curvature plot is continuous and consists of only a few monotone pieces
A fair curve should only have curvature extrema where the designer explicitly wishes so
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Our focus
• To satisfy the fairness conditions we use G2 splines, that consist of spiral segments
• A spiral is a curved line segment whose curvature varies monotonically with arc-length
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Designer control
An intuitive way to control our curves is required
Use hiearchy of circles:
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Designer control – an alternative
• If we let the user specify the curve segment’s starting- and endpoints on the control circles, we can formulate the problem as geometric Hermite interpolation
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Designer control – an alternative
• Given are position, tangent, and curvature data at each the endpoint
• Find a Bézier curve which reconstructs these quantities at it’s endpoints
• These are 2x4 scalar constraints on each segment
Position: 2 scalarTangent: 1 scalarCurvature: 1 scalar
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Designer control – an alternative Cubic Bézier solution for GH
• A cubic Bézier curve has 8 scalar degrees of freedom
• A depressed quartic equation results from
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Designer control – an alternative Cubic Bézier solution for GH
• With appropriate geometric constraints on position, tangent and curvature the following system has positive real roots
• No spiral (cubic Bézier spirals have 5 degrees of freedom)
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Designer control
• Let us use Pythagorean Hodograph spirals for the transition curves
curve curve’s hodograph
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Let the parameterization be such that
For some integral polynomial
The arc-length
can be expressed in closed-form
Pythagorean Hodographs
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Theorem: the Pythagorean condition
for polynomials holds if and only if they can be expressed in terms of other polynomials as
where u(t) and v(t) are relatively prime.
Pythagorean Hodographs
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Pythagorean Hodographs
PH curves’ hodographs satisfy:
PH curves of degree n have n+3 degrees of freedom
General polynomials of degree n have 2n + 2 degrees of freedom
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Pythagorean Hodographs
Arc-length is a polynomial
Offset of a degree n PH curve is a rational polynomial of degree 2n-1
For practical usage Cubic PH curves cannot have an inflection point
We use quintic PH curves We need to find u(t), v(t) quadratic polynomials in
Bézier form
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Method
• Let the user create a hierarchy of control circles
• Create spiral segments between a node and its descendants– Three cases are possible:
• Circles can be connected by an S-shaped circle-to-circle curve
• Circles can be connected by a C-shaped circle-in-circle curve
• The circles cannot be connected
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Circle-to-circle transition
• The circles have to be non-touching and non-overlapping
• We used the work of Walton and Meek to define the quintic PH curve’s control points
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Circle-to-circle transition
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Circle-to-circle transition
• The circle centres have to be within a certain distance (depending on their radii)
• We have to solve
• Where
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Circle-in-circle transition
• A fully contained circle is joined to its ancestor if such transition is possible
• The conditions and the derivation of control points can be found in Habib and Sakai’s work
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Circle-in-circle transition
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Circle-in-circle transition
• Constraints on the radius of the smaller circle and its distance from the big circle
• In our tool the user only specifies that a circle is needed within a given control circle, it’s position and radius will be computed automatically
• The resulting smaller circle can be adjusted within the valid range of solutions
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Export
• Since most vector graphics systems support cubic Bézier curve’s we provide export in such format
• The quintic Bézier curve is approximated by cubic Bézier segments
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Design process
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Future work
• Integration into vector graphics systems
• More streamlined workflow
• Use of improved transition curves
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Pythagorean Hodographs
Theorem (Farouki): It is not possible to parameterize any plane curve, other than a straight line, by rational functions of its arc length.
No unit-speed parameterizationArc-length in closed form:
Degree 2: possible with logarithmic termsDegree 3: possible, but only with elliptic functions
Let us use a subset of polynomial curves that has closed form for arc-length
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Pottmann introduced the RPRO set of curves
The rational curve is obtained as the envelope of its tangent line:
Where h(t) is the signed distance of the tangent line from the origin and n(t) is a rational unit normal vector to the tangent line g(t) given by
Rational polynomials with rational offsets
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Rational polynomials with rational offsets
The envelope of g(t) family can be found by solving a linear system for g(t) and g’(t) for x and y as a function of t
This subset of rational polynomials is closed under offsetting
The offset curve’s degree is the same as that of the curve to be offset
Arc-length is not a rational function, in general
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Rational polynomials with rational offsets
Evolute:
where
Theorem: the rational curves c(t), whose arc-length parameter s(t) is a rational function of t, are exactly the rational offsets of the RPRO curves’ evolutes.
The evolutes of polynomial or rational polynomial curves are always rational polynomials
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Previous work
• Definition of “fair” is not obvious:– Farin: "A curve is fair if its curvature plot is
continuous and consists of only a few monotone pieces."
• A spiral's curvature varies monotonically with arc-length
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Ábrák
Circle to circle S C
Circle in circle Valami egész ábrás bigyó Esetleg egy visszafogott, oldalra kirakandó
ábra, pl. sarokba vmi növény?
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Bevezetés Miről lesz szó, nagyjából a poszter lényege
Motiváció, korábbi munkák A cikkek amiket küldtél meg még?
Saját munka: Kör hierarchia Körök közti különböző átmenetek Gyakorlati megfontolások (köbös Bézier konverzió meg
hasonlók) További teendők
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Our goal
In general: – creation of aesthatically pleasing curves
In particular: – smoothly curving design element creation
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Our goal
Transition curves between two circular arcs For design of highways: G2 with few
curvature extrema Farin: "A curve is fair if its curvature plot is
continuous and consists of only a few monotone pieces."
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Introduction and background• Pythagorean hodograph• curves with no curvature extrema for an S-shaped transition
• single curvature extremum for a C-shaped transition• are suitable for the design of fair curves
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Motivation
Ezek voltak: ...