Back

Free-form deformation of solid geometric models

Thomas W. SederbergScott R. Parry

article1986SIGGRAPH3,353 citations

Introduces free-form deformation, a foundational technique that sculpts arbitrary solid geometric models, parametric surfaces, and polygonal meshes by embedding them in a lattice of trivariate Bernstein polynomial control points with derivative continuity and volume-preservation controls.

Listen

A new technique called free-form deformation enables designers to reshape solid geometric models by embedding them in a flexible lattice and then moving control points on that lattice, much as a sculptor works with clay. Solid modeling systems have long relied on simple surfaces such as planes and quadrics, while free-form surface modeling has remained largely separate; the gap has made it difficult to create and edit complex, curved solids inside standard modeling pipelines.

The authors set out to show that a single, simple mapping based on trivariate Bernstein polynomials can deform any solid model—whether defined by constructive solid geometry or boundary representation—while preserving the validity of the underlying representation. They formulated the deformation by first attaching a local coordinate frame to a parallelepiped region, placing a regular grid of control points inside it, and then displacing those points; the new position of any point inside the solid is obtained by evaluating the resulting tensor-product polynomial. The method was demonstrated on polygonal data, implicit surfaces, parametric patches, and complete CSG models, including hierarchical sequences of local and global deformations.

The work shows that free-form deformation applies uniformly to every common surface type and modeling scheme, that derivative continuity across patch boundaries can be maintained by leaving neighboring control points fixed, and that a family of deformations leaves volume exactly unchanged. It further demonstrates that parametric curves and surfaces remain parametric after deformation, that mass-property calculations stay feasible, and that the technique produces usable results for aesthetic shapes, transition surfaces, and some blended features. Professional stylists who tried the method found the sculpting metaphor natural and efficient; one complete telephone handset model was produced in a single design pass after only a few hours of practice.

These capabilities matter because they let engineers and stylists edit free-form solids inside existing solid-modeling systems without switching representations or losing downstream analysis tools. Volume preservation removes a common source of error in packaging and fluid-volume calculations, while the ability to keep surfaces parametric supports continued use of standard intersection and rendering algorithms.

The authors note three practical limits: the method does not yet provide general filleting and blending, local deformations produce planar interfaces with the undeformed region, and operations on the trivariate polynomials are more expensive than the corresponding surface operations. They also observe that the full usefulness of volume-preserving deformations remains to be explored. Further work on display speed, inverse mapping robustness, and integration with commercial fillet routines would strengthen the approach before broad deployment.

Brigham Young UniversityMilliken & Co.elastic deformationgeometric modelingFree‑Form DeformationSolid Geometric ModelTrivariate Bernstein Polynomial
Sederberg et al (1986).pdf

No sufficiently relevant recommendations were found.

  • Paper: Lucas-Kanade 20 Years On: A Unifying Framework, Simon Baker et al. (2004). This paper builds directly upon foundational continuous spatial transformations and optimization techniques established in early graphics and vision literature to unify subsequent algorithmic variants.
Cover for Free-form deformation of solid geometric models

Abstract

A technique is presented for deforming solid geometric models in a free-form manner. The technique can be used with any solid modeling system, such as CSG or B-rep. It can deform surface primitives of any type or degree: planes, quadrics, parametric surface patches, or implicitly defined surfaces, for example. The deformation can be applied either globally or locally. Local deforma- tions can be imposed with any desired degree of deriva- tive continuity. It is also possible to deform a solid model in such a way that its volume is preserved.

The scheme is based on trivariate Bernstein polyno- mials, and provides the designer with an intuitive appre- ciation for its effects.

Topics

elastic deformationgeometric modeling

Keywords

Free‑Form DeformationSolid Geometric ModelTrivariate Bernstein PolynomialPreserve VolumeApply Local DeformationMaintain Derivative ContinuityUse CSGModel Implicit SurfaceRepresent Parametric SurfaceControl Point Lattice

Table of Contents

  • 1. INTRODUCTION
  • 2. FORMULATING FREE-FORM DEFORMATIONS
  • 2.1. Deformation Domain
  • 3. CONTINUITY CONTROL
  • 3.1. Local Deformations
  • 4. VOLUME CHANGE
  • 5. APPLICATIONS
  • 6. CONCLUSIONS

Knowls

  1. Knowl 1 — Trivariate Bernstein Free-Form Deformation Mapping

    model/method

    Free-Form Deformation (FFD) embeds an arbitrary geometric model into a flexible parallelepiped region defined by an origin X0∈R3\mathbf{X}_0 \in \mathbb{R}^3 and three spanning vectors S,T,U∈R3\mathbf{S}, \mathbf{T}, \mathbf{U} \in \mathbb{R}^3. A grid of control points Pijk\mathbf{P}_{ijk} is imposed on the lattice, forming l+1l+1 planes in the S\mathbf{S} direction, m+1m+1 planes in the T\mathbf{T} direction, and n+1n+1 planes in the U\mathbf{U} direction. In their initial undeformed configuration, the control points lie at uniform lattice coordinates:

    Pijk=X0+ilS+jmT+knU,i∈{0,…,l},  j∈{0,…,m},  k∈{0,…,n}.\mathbf{P}_{ijk} = \mathbf{X}_0 + \frac{i}{l}\mathbf{S} + \frac{j}{m}\mathbf{T} + \frac{k}{n}\mathbf{U}, \quad i \in \{0,\dots,l\},\; j \in \{0,\dots,m\},\; k \in \{0,\dots,n\}.

    A user specifies a deformation by displacing the control points Pijk\mathbf{P}_{ijk} from their initial grid positions. For any point with local coordinates (s,t,u)∈[0,1]3(s,t,u) \in [0,1]^3, the displaced Cartesian coordinates Xffd(s,t,u)\mathbf{X}_{\text{ffd}}(s,t,u) are evaluated using a tensor-product trivariate Bernstein polynomial:

    Xffd(s,t,u)=∑i=0l(li)(1−s)l−isi∑j=0m(mj)(1−t)m−jtj∑k=0n(nk)(1−u)n−kukPijk.\mathbf{X}_{\text{ffd}}(s,t,u) = \sum_{i=0}^l \binom{l}{i}(1-s)^{l-i}s^i \sum_{j=0}^m \binom{m}{j}(1-t)^{m-j}t^j \sum_{k=0}^n \binom{n}{k}(1-u)^{n-k}u^k \mathbf{P}_{ijk}.

    Under this transformation, the 12 straight edges of the bounding parallelepiped deform into Bézier curves, and its 6 planar boundary faces deform into tensor-product Bézier surface patches.

  2. Knowl 2 — Local Coordinate Computation for Parallelepiped Lattice

    equation

    For a local coordinate system defined on a parallelepiped volume with origin X0∈R3\mathbf{X}_0 \in \mathbb{R}^3 and three linearly independent axis vectors S,T,U∈R3\mathbf{S}, \mathbf{T}, \mathbf{U} \in \mathbb{R}^3, the Cartesian coordinates of any point X∈R3\mathbf{X} \in \mathbb{R}^3 are expressed as:

    X=X0+sS+tT+uU.\mathbf{X} = \mathbf{X}_0 + s\mathbf{S} + t\mathbf{T} + u\mathbf{U}.

    The local (s,t,u)(s,t,u) coordinates are determined analytically via scalar triple products:

    s=(T×U)⋅(X−X0)(T×U)⋅S,t=(S×U)⋅(X−X0)(S×U)⋅T,u=(S×T)⋅(X−X0)(S×T)⋅U.s = \frac{(\mathbf{T} \times \mathbf{U}) \cdot (\mathbf{X} - \mathbf{X}_0)}{(\mathbf{T} \times \mathbf{U}) \cdot \mathbf{S}}, \quad t = \frac{(\mathbf{S} \times \mathbf{U}) \cdot (\mathbf{X} - \mathbf{X}_0)}{(\mathbf{S} \times \mathbf{U}) \cdot \mathbf{T}}, \quad u = \frac{(\mathbf{S} \times \mathbf{T}) \cdot (\mathbf{X} - \mathbf{X}_0)}{(\mathbf{S} \times \mathbf{T}) \cdot \mathbf{U}}.

    For any point strictly interior to the parallelepiped lattice, the coordinates satisfy 0<s<10 < s < 1, 0<t<10 < t < 1, and 0<u<10 < u < 1.

  3. Knowl 3 — Preservation of Parametric Representation under FFD

    theoretical result

    When a parametric surface X(α,β)=(f(α,β),g(α,β),h(α,β))\mathbf{X}(\alpha, \beta) = (f(\alpha, \beta), g(\alpha, \beta), h(\alpha, \beta)) undergoes Free-Form Deformation Xffd(x,y,z)\mathbf{X}_{\text{ffd}}(x, y, z), the deformed surface patch Xffd(α,β)\mathbf{X}_{\text{ffd}}(\alpha, \beta) is obtained directly through functional composition:

    Xffd(α,β)=Xffd(f(α,β),g(α,β),h(α,β)).\mathbf{X}_{\text{ffd}}(\alpha, \beta) = \mathbf{X}_{\text{ffd}}(f(\alpha, \beta), g(\alpha, \beta), h(\alpha, \beta)).

    Because the composition of polynomial or rational functions with trivariate Bernstein polynomials yields polynomial or rational expressions, the deformed entity remains an exact parametric surface without altering the underlying surface topology or requiring polygon approximation. As a corollary, parametric curves X(α)\mathbf{X}(\alpha) (including intersection curves between deformed primitives) also remain exact parametric curves under FFD.

  4. Knowl 4 — Cross-Boundary Continuity Conditions between Adjacent FFDs

    theoretical result

    Let two adjacent FFD volumes X1(s1,t1,u1)\mathbf{X}_1(s_1, t_1, u_1) and X2(s2,t2,u2)\mathbf{X}_2(s_2, t_2, u_2) share an interface at s1=s2=0s_1 = s_2 = 0. For an embedded surface defined locally by parameters (v,w)(v,w) such that (s,t,u)=(s(v,w),t(v,w),u(v,w))(s,t,u) = (s(v,w), t(v,w), u(v,w)), the partial derivatives of the deformed surface are given by the chain rule:

    ∂X∂v=∂X∂s∂s∂v+∂X∂t∂t∂v+∂X∂u∂u∂v,∂X∂w=∂X∂s∂s∂w+∂X∂t∂t∂w+∂X∂u∂u∂w.\frac{\partial \mathbf{X}}{\partial v} = \frac{\partial \mathbf{X}}{\partial s}\frac{\partial s}{\partial v} + \frac{\partial \mathbf{X}}{\partial t}\frac{\partial t}{\partial v} + \frac{\partial \mathbf{X}}{\partial u}\frac{\partial u}{\partial v}, \quad \frac{\partial \mathbf{X}}{\partial w} = \frac{\partial \mathbf{X}}{\partial s}\frac{\partial s}{\partial w} + \frac{\partial \mathbf{X}}{\partial t}\frac{\partial t}{\partial w} + \frac{\partial \mathbf{X}}{\partial u}\frac{\partial u}{\partial w}.

    Because the geometric derivatives ∂s∂v,∂t∂v,∂u∂v,∂s∂w,∂t∂w,∂u∂w\frac{\partial s}{\partial v}, \frac{\partial t}{\partial v}, \frac{\partial u}{\partial v}, \frac{\partial s}{\partial w}, \frac{\partial t}{\partial w}, \frac{\partial u}{\partial w} depend only on the undeformed model, sufficient conditions for C1C^1 continuity across the interface are:

    ∂X1(0,t,u)∂s=∂X2(0,t,u)∂s,∂X1(0,t,u)∂t=∂X2(0,t,u)∂t,∂X1(0,t,u)∂u=∂X2(0,t,u)∂u.\frac{\partial \mathbf{X}_1(0,t,u)}{\partial s} = \frac{\partial \mathbf{X}_2(0,t,u)}{\partial s}, \quad \frac{\partial \mathbf{X}_1(0,t,u)}{\partial t} = \frac{\partial \mathbf{X}_2(0,t,u)}{\partial t}, \quad \frac{\partial \mathbf{X}_1(0,t,u)}{\partial u} = \frac{\partial \mathbf{X}_2(0,t,u)}{\partial u}.

    C0C^0 continuity holds if the shared control point plane remains coincident. Higher CkC^k derivative continuity is achieved by enforcing standard Bézier volume cross-boundary continuity conditions on the control lattices.

  5. Knowl 5 — $C^k$ Continuity Conditions for Local Free-Form Deformations

    theoretical result

    When a local FFD is applied to a subregion of a solid model, the undeformed surrounding model can be treated as an adjacent, undisplaced FFD lattice. A sufficient condition for ensuring CkC^k derivative continuity across the planar interface where the FFD boundary face intersects the geometric model is that the control points on the kk planes of the lattice adjacent to and including the interface plane remain fixed in their initial, undisplaced positions.

    This condition can be independently enforced across any face of the FFD lattice intersected by the model to guarantee smooth blending between the locally deformed region and undeformed regions.

  6. Knowl 6 — Volume Change Bounds via Bernstein Jacobian Coefficients

    theoretical result

    When a solid model undergoes a deformation mapping F(x,y,z)=(F(x,y,z),G(x,y,z),H(x,y,z))\mathbf{F}(x,y,z) = (F(x,y,z), G(x,y,z), H(x,y,z)), the volume of any differential volume element dx dy dzdx\,dy\,dz transforms into Jac⁡(F(x,y,z)) dx dy dz\operatorname{Jac}(\mathbf{F}(x,y,z))\,dx\,dy\,dz, where the Jacobian determinant is:

    Jac⁡(F)=det⁡(∂F∂x∂F∂y∂F∂z∂G∂x∂G∂y∂G∂z∂H∂x∂H∂y∂H∂z).\operatorname{Jac}(\mathbf{F}) = \det \begin{pmatrix} \frac{\partial F}{\partial x} & \frac{\partial F}{\partial y} & \frac{\partial F}{\partial z} \\[6pt] \frac{\partial G}{\partial x} & \frac{\partial G}{\partial y} & \frac{\partial G}{\partial z} \\[6pt] \frac{\partial H}{\partial x} & \frac{\partial H}{\partial y} & \frac{\partial H}{\partial z} \end{pmatrix}.

    The deformed volume is the triple integral of Jac⁡(F)\operatorname{Jac}(\mathbf{F}) over the undeformed solid domain. When Jac⁡(F)\operatorname{Jac}(\mathbf{F}) is expressed as a trivariate Bernstein polynomial, the convex hull property of Bernstein polynomials guarantees that the minimum and maximum polynomial coefficients provide strict lower and upper bounds, respectively, on the local volume change factor throughout the deformation region.

  7. Knowl 7 — Volume-Preserving Free-Form Deformations

    theoretical result

    A Free-Form Deformation mapping F(x,y,z)\mathbf{F}(x,y,z) is strictly volume preserving if its Jacobian determinant satisfies:

    Jac⁡(F(x,y,z))≡1\operatorname{Jac}(\mathbf{F}(x,y,z)) \equiv 1

    everywhere throughout the deformation domain. Any solid geometric model deformed under such a transformation retains exactly its original volume.

  8. Knowl 8 — Point Classification on Deformed Implicit Models via Inverse FFD Mapping

    model/method

    For solid models defined by implicit equations or Constructive Solid Geometry (CSG) primitives, point classification (determining if a query Cartesian point Xcart\mathbf{X}_{\text{cart}} is inside, outside, or on the surface) requires solving the inverse mapping Xffd(s,t,u)=Xcart\mathbf{X}_{\text{ffd}}(s,t,u) = \mathbf{X}_{\text{cart}} for the local coordinates (s,t,u)(s,t,u).

    The inverse mapping is computed numerically using trivariate Newton-Raphson iteration or control lattice spatial subdivision. Once (s,t,u)(s,t,u) are obtained, the corresponding undeformed coordinates X=X0+sS+tT+uU\mathbf{X} = \mathbf{X}_0 + s\mathbf{S} + t\mathbf{T} + u\mathbf{U} are evaluated in the implicit surface equation f(X)=0f(\mathbf{X}) = 0. Inversion may become ill-conditioned or multi-valued if the Jacobian Jac⁡(F)\operatorname{Jac}(\mathbf{F}) changes sign within the domain.

  9. Knowl 9 — Limitations of Free-Form Deformation

    limitation

    Free-Form Deformation has three principal limitations:

    1. It cannot generate general filleting and blending surfaces between intersecting objects.
    2. The boundary between a local FFD region and undeformed regions of the model is constrained to be planar (coinciding with the faces of the FFD lattice). Producing an arbitrary curved deformation boundary requires pre-deforming the initial lattice, which is computationally expensive.
    3. Numerical and symbolic operations on trivariate Bernstein polynomials (such as lattice subdivision, degree elevation, and root finding/inversion) carry substantially higher computational cost than corresponding bivariate surface patch operations.

Coverage note — Specific qualitative demonstration models (telephone handset, trophy handle, and fairing ducts) were omitted as standalone knowls because their underlying principles are fully represented in the method, continuity, and property knowls.

References

  1. 1.Barr, A. H. Superquadrics and angle-preserving transformations, IEEE Computer Graphics and Applications, 1 (January 1981), 11-23.
  2. 2.Barr, A. H. Global and Local Deformations of Solid Primitives, Computer Graphics 17,3 (July 1984), pp 21-30.
  3. 3.Bézier, P. Mathematical and practical possibilities of UNISURF, in Computer Aided Geometric Design, R. E. Barnhill and R. F. Riesenfeld, eds., Academic Press (1974), New York, pp 127-152.
  4. 4.Blinn, J. A generalization of algebraic surface drawing, ACM Transactions on Graphics, 1 (July 1982), 235-256.
  5. 5.Boehm, W., Farin, G. and Kahmann, J. A survey of curve and surface methods in CAGD. Computer Aided Geometric Design 1,1(July 1984), 1-60.
  6. 6.Brewer, J. A. and Anderson, D. C. Visual interaction with Overhauser curves and surfaces. Computer Graphics 11, 2 (July 1977), 132-137.
  7. 7.Casale, M.S. and Stanton, E. L., An overview of analytic solid modeling, IEEE Computer Graphics and Applications, 5 (February 1985), 45-56.
  8. 8.Chiyokura, H. and Kimura, F., Design of solids with free-form surfaces, Computer Graphics, 17 (July 1983), 289-298.
  9. 9.Cobb, E. S. Design of Sculptured Surfaces using the B-spline Representation. Ph.D. Dissertation, Department of Computer Science, University of Utah, June 1984.
  10. 10.Farouki, R. T. and Hinds, J. K., A hierarchy of geometric forms, IEEE Computer Graphics and Applications, 5 (May 1985), 51-78.
  11. 11.Hoffmann, C. and Hopcroft, J. Automatic surface generation in Computer Aided Design. TR 85-661, Dept. of Computer Science, Cornell University, January 1985.
  12. 12.Jared, G. E. M., Synthesis of volume modeling and sculptured surfaces in BUILD, CAD84, Computers in Design Engineering Conference Proceedings, (1984), 481-495.
  13. 13.Kalay, Y. E., Modeling polyhedral solids bounded by multi-curved parametric surfaces, ACM IEEE Nineteenth Design Automation Conference Proceedings (June 1982), 501-507.
  14. 14.Kimura, F., Geomap-III: Designing solids with free-form surfaces, IEEE Computer Graphics and Applications, 4 (1984), 58-72.
  15. 15.Middleditch, A. E. and Sears, K. H. Blend surfaces for set theoretic volume modelling systems. Computer Graphics 19, 3 (July 1985), 161-170.
  16. 16.Owen, J. and Rockwood, A. P., General implicit surfaces in geometric modeling, in Geometric Modeling, G. Farin, editor, SIAM, to appear (1986).
  17. 17.Parent, R. E. A system for sculpting 3-D data. Computer Graphics 11, 2 (July 1977), 138-147.
  18. 18.Parry, S. R. Free-form deformations in a constructive solid geometry modeling system, Ph.D. Dissertation, Department of Civil Engineering, Brigham Young University, April 1986.
  19. 19.Requicha, A. A. G. and Voelcker, H. B. Solid modeling: A historical summary and contemporary assessment. IEEE Computer Graphics and Applications 2,2(March 1982), 9-24.
  20. 20.Ricci, A., A constructive geometry for computer graphics, Computer Journal 16 (1973), 157-160.
  21. 21.Riesenfeld, R. F., A view of spline-based solid modelling, Proceedings, Autofac V, Detroit, MI (November 1983), 75-83.
  22. 22.Rockwood, A. P. and Owen, J. Blending surfaces in solid modeling, in Geometric Modeling, G. Farin, editor, SIAM, to appear (1986).
  23. 23.Sabin, M. A. The use of potential surfaces in numerical control, British Aircraft Corporation, Weybridge VTO/MS/153 (1968).
  24. 24.Sabin, M. A. Interrogation techniques for parametric surfaces, Proceedings, Computer Graphics '70, Brunel University, April 1970.
  25. 25.Sarraga, R. F and Waters, W. C., Free-form surfaces in GMSolid: Goals and issues, in Solid Modeling by Computers from Theory to Applications, M. S. Pickett and J. W. Boyse, editors, Plenum Press (1984), 187-204.
  26. 26.Sederberg, T. W. and Parry, S. R., Free-form deformation of polygonal data, Proceedings, International Electronic Image Week, Nice, France (April 1986), 633-639.
  27. 27.Sederberg, T. W. and Ferguson, H. R. P. Volume preserving deformations. unpublished notes (1986b).
  28. 28.Stanton, E. L., Crain, L. M. and Neu, T. F., A parametric cubic modeling system for general solids of composite material, International Journal of Numerical Methods in Engineering, 11 (1977), 653-670.
  29. 29.Steinberg, H. A., A smooth surface based on biquadratic patches, IEEE Computer Graphics and Applications, 4 (September 1984), 20-23.
  30. 30.Thomas, S. W., Modeling volumes bounded by B-spline surfaces, Ph.D. Thesis, Dept. of Computer Science, University of Utah, (1984).
  31. 31.Varady, T. and Pratt, M. J. Design techniques for the definition of solid objects with free-form geometry. Computer Aided Geometric Design 1,3 (Dec. 1984), 207-225.
  32. 32.Veenman, P. R., The design of sculptured surfaces using recursive subdivision techniques, in: Proc. Conf. on CAD/CAM Technology in Mechanical Engineering, MIT, Cambridge (March 1982).

Citation

MLA
Sederberg, T. W., and S. R. Parry. “Free-form Deformation of Solid Geometric Models”. Proceedings of the 13th Annual Conference on Computer Graphics and Interactive Techniques, 1986, pp. 151–60, https://doi.org/10.1145/15922.15903.
APA
Sederberg, T. W., & Parry, S. R. (1986). Free-form deformation of solid geometric models. Proceedings of the 13th Annual Conference on Computer Graphics and Interactive Techniques, 151–160. https://doi.org/10.1145/15922.15903
Chicago
Sederberg, T. W., and S. R. Parry. 1986. “Free-form Deformation of Solid Geometric Models”. Proceedings of the 13th Annual Conference on Computer Graphics and Interactive Techniques, 151–60. https://doi.org/10.1145/15922.15903.
Harvard
Sederberg, T.W. and Parry, S.R. (1986) “Free-form deformation of solid geometric models”, Proceedings of the 13th annual conference on Computer graphics and interactive techniques. ACM, pp. 151–160. Available at: https://doi.org/10.1145/15922.15903.
Vancouver
1. Sederberg TW, Parry SR (1986) Free-form deformation of solid geometric models. In: Proceedings of the 13th annual conference on Computer graphics and interactive techniques. ACM, pp 151–160

BibTeX

@inproceedings{Sederberg_1986, series={SIGGRAPH ’86}, title={Free-form deformation of solid geometric models}, url={http://dx.doi.org/10.1145/15922.15903}, DOI={10.1145/15922.15903}, booktitle={Proceedings of the 13th annual conference on Computer graphics and interactive techniques}, publisher={ACM}, author={Sederberg, Thomas W. and Parry, Scott R.}, year={1986}, month=Aug, pages={151–160}, collection={SIGGRAPH ’86} }
Metadata:Crossref

Access the Paper

This paper is available from its original source. Click below to access the PDF.

Open PDF