Auxiliary Functions

is_rigid(F[; tol, tol, tested_random_flexes, symmetric_newton])

Heuristically checks if a geometric constraint system F is (continuously) rigid.

See DeformationPath(G::DeformationPaths.ConstraintSystem, motion_samples::Vector{<:Vector{<:Real}}) for a description of the possible parameters.

is_inf_rigid(F[; tol_rank_drop])

Checks if a geometric constraint system F is infinitesimally rigid.

is_prestress_stable(F)

Checks if a geometric constraint system F is prestress stable.

is_second_order_rigid(F)

Checks if a geometric constraint system F is second-order rigid.

See also compute_nonblocked_flex for the possible keywords.

euler_step(G, step_size, prev_flex, point, K_n)

Euler step predicting the next point along the approximate motion.

Returns

• predicted_point::Vector{<:Real}: The next point predicted by Euler's method.
• predicted_inf_flex::Vector{<:Real}: The tangent vector predicted by Euler's method.

newton_correct(G, point)

Apply Newton's method to correct point back to the constraints in G.

Arguments

• G::ConstraintSystem: The underlying geometric constraint system.
• point::Vector{<:Real}: The initial point that Newton's method is applied to.

For further keywords, see newton_correct.

Returns

• q::Vector{<:Real}: A point q such that the Euclidean norm of the evaluated equations is at most tol

newton_correct(equations, variables, jac, point[; tol, time_penalty])

Apply Newton's method to correct point back to the constraints in equations.

Arguments

• equations::Vector{Expression}: Equations to correct point to.
• variables::Vector{Variable}: Variables from the affine coordinate ring.
• jac::Matrix{Expression}: Jacobian matrix corresponding to equations and variables.
• point::Vector{<:Real}: The initial point that Newton's method is applied to.
• tol::Real (optional): Numerical tolerance that is used as a stopping criterion for Newton's method. Default value: 1e-13.
• time_penalty::Union{Real,Nothing} (optional): If Newton's method takes too long, we stop the iteration and throw an error. Here, "too long" is measured in terms of length(point)/time_penalty seconds. Default value: 2.
• armijo_linesearch::Bool (optional): Flag for activating the Armijo backtracking line search procedure. Default value: true.

Returns

• q::Vector{<:Real}: A point q such that the Euclidean norm of the evaluated equations is at most tol

symmetric_newton_correct(G, point[; tol, time_penalty])

Apply symmetric Newton's method to correct point back to the constraints in equations.

The symmetric Newton corrector evaluates the Jacobian matrix less often.

Arguments

• G::ConstraintSystem: The underlying geometric constraint system.
• point::Vector{<:Real}: The initial point that Newton's method is applied to.
• tol::Real (optional): Numerical tolerance that is used as a stopping criterion for Newton's method. Default value: 1e-13.
• time_penalty::Union{Real,Nothing} (optional): If Newton's method takes too long, we stop the iteration and throw an error. Here, "too long" is measured in terms of length(point)/time_penalty seconds. Default value: 2.

Returns

• q::Vector{<:Real}: A point q such that the Euclidean norm of the evaluated equations is at most tol

See also symmetric_newton_correct

symmetric_newton_correct(equations, variables, jac, point[; tol, time_penalty])

Apply symmetric Newton's method to correct point back to the constraints in equations.

The symmetric Newton corrector evaluates the Jacobian matrix less often.

Arguments

• equations::Vector{Expression}: Equations to correct point to.
• variables::Vector{Variable}: Variables from the affine coordinate ring.
• jac::Matrix{Expression}: Jacobian matrix corresponding to equations and variables.
• point::Vector{<:Real}: The initial point that Newton's method is applied to.
• tol::Real (optional): Numerical tolerance that is used as a stopping criterion for Newton's method. Default value: 1e-13.
• time_penalty::Union{Real,Nothing} (optional): If Newton's method takes too long, we stop the iteration and throw an error. Here, "too long" is measured in terms of length(point)/time_penalty seconds. Default value: 2.

Returns

• q::Vector{<:Real}: A point q such that the Euclidean norm of the evaluated equations is at most tol

compute_inf_flexes(G, point[; tol])

Compute all infinitesimal flexes of a geometric constraint system G in point.

compute_equilibrium_stresses(G, point[; tol])

Compute all equilibrium stresses of a geometric constraint system G in point.

compute_nontrivial_inf_flexes(G, point, K_n[; tol])

Compute the nontrivial infinitesimal flexes of a geometric constraint system G in point.

compute_nonblocked_flex(F[; tol_rank_drop, tol])

Compute an infinitesimal flex of F that is not blocked by an equilibrium stress.

to_Array(G, p)

Transform a realization p to a vector of coordinates.

to_Array(F, p)

Transform a realization p to a vector of coordinates.

to_Array(F, p)

Transform a realization p to a vector of coordinates.

to_Matrix(G, q)

Transform a vector of coordinates q to a realization matrix.

to_Matrix(F, q)

Transform a vector of coordinates q to a realization matrix.

minors(A, k)

Compute the (k+1)x(k+1) minors of the matrix A

fix_antipodals!(F)

Entangles the antipodal points in a polytope F so that their position is constrained to antipodal points on a sphere.

tetrahedral_symmetry!(F)

Add constraints to a polytope F representing a tetrahedral symmetry.

triangle_shrinking(F)

Evenly shrink the triangular facets of a given polytope and compute the nontrivial infinitesimal flexes in each step.