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@@ -16,7 +16,7 @@ with the performance and versatility of C++. The tutorial is a self-contained,
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hands-on introduction to libigl. Via live coding and interactive examples, we
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hands-on introduction to libigl. Via live coding and interactive examples, we
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demonstrate how to accomplish various common geometry processing tasks such as
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demonstrate how to accomplish various common geometry processing tasks such as
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computation of differential quantities and operators, real-time deformation,
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computation of differential quantities and operators, real-time deformation,
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-global parametrization, numerical optimization and mesh repair. Each section
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+global parametrization, numerical optimization and mesh repair. Each section
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of these lecture notes links to a cross-platform example application.
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of these lecture notes links to a cross-platform example application.
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# Table of Contents
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# Table of Contents
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@@ -792,13 +792,13 @@ deformation.
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### Biharmonic deformation fields
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### Biharmonic deformation fields
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Now we know that one useful property for a deformation technique is "rest pose
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Now we know that one useful property for a deformation technique is "rest pose
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reproduction": applying no deformation to the handles should apply no
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reproduction": applying no deformation to the handles should apply no
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-deformation to the shape.
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+deformation to the shape.
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To guarantee this by construction we can work with _deformation fields_ (ie.
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To guarantee this by construction we can work with _deformation fields_ (ie.
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displacements)
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displacements)
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-$\mathbf{d}$ rather
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+$\mathbf{d}$ rather
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than directly with positions $\mathbf{x}. Then the deformed positions can be
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than directly with positions $\mathbf{x}. Then the deformed positions can be
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-recovered as
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+recovered as
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$\mathbf{x}' = \mathbf{x}+\mathbf{d}.$
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$\mathbf{x}' = \mathbf{x}+\mathbf{d}.$
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@@ -907,7 +907,7 @@ as a constrained optimization problem [#jacobson_2011][]. The weights enforce
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smoothness by minimizing a smoothness energy: the familiar Laplacian energy:
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smoothness by minimizing a smoothness energy: the familiar Laplacian energy:
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$\sum\limits_{i = 1}^m \int_S (\Delta w_i)^2 dA$
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$\sum\limits_{i = 1}^m \int_S (\Delta w_i)^2 dA$
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-
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+
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subject to constraints which enforce interpolation of handle constraints:
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subject to constraints which enforce interpolation of handle constraints:
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$w_i(\mathbf{x}) = \begin{cases} 1 & \text{ if } \mathbf{x} \in H_i\\ 0 & \text{ otherwise }
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$w_i(\mathbf{x}) = \begin{cases} 1 & \text{ if } \mathbf{x} \in H_i\\ 0 & \text{ otherwise }
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