Support for the real projective plane #1405
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geometry/geometry.vcxproj
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| @@ -292,6 +292,8 @@ | |||
| <ClInclude Include="named_quantities.hpp" /> | |||
| <ClInclude Include="pair.hpp" /> | |||
| <ClInclude Include="pair_body.hpp" /> | |||
| <ClInclude Include="perspective.hpp" /> | |||
| <ClInclude Include="perspective_body.hpp" /> | |||
geometry/r3_element.hpp
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| struct enable_if_normed<R3Element<T, T, T>> : not_constructible { | ||
| using Scalar = T; | ||
| }; | ||
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Add a type alias (norm_type<T> or somesuch) for typename enable_if_normed<T>::Scalar, and use it below.
geometry/r3_projective.hpp
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| // the points at infinity. Note that this identification is not unique, so | ||
| // the comparison of the values returned by this function is not an | ||
| // equivalence on the projective plane. | ||
| double point_at_infinity() const; |
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Document that this must be at infinity.
geometry/r3_projective.hpp
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| // equivalence on the projective plane. | ||
| double point_at_infinity() const; | ||
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| // Returns the Euclidean (non-homegeneous) coordinates of the point. May be |
geometry/r3_projective_body.hpp
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| R3Projective<Scalar>::R3Projective( | ||
| R3Element<Scalar, Scalar, double> const& coordinates) | ||
| : coordinates_(coordinates) { | ||
| // (0, 0, 0) does not represent any point. |
geometry/r3_projective_body.hpp
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| template<typename Scalar> | ||
| std::string DebugString(R3Projective<Scalar> const & r3_projective) { | ||
| return DebugString(r3_projective.coordinates_); |
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Let's use the notation [x:y:z], it is fairly unambiguous (which is rare enough), and it will allow us to distinguish homogeneous coordinates from vectors. See https://en.wikipedia.org/wiki/Projective_space#Projective_space_as_a_manifold.
geometry/r3_projective.hpp
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| // must be a 1-dimensional vector space over ℝ, typically represented by | ||
| // |Quantity| or |double|. | ||
| template<typename Scalar> | ||
| class R3Projective { |
geometry/r3_element.hpp
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| // is also normed. | ||
| // |R3Element| is the underlying data type for more advanced strongly typed | ||
| // structures such as |Multivector|. | ||
| template<typename ScalarX, typename ScalarY, typename ScalarZ> |
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As discussed, let's not go there, we're not using the vector space structure in the projective plane.
geometry/rp2_element_body.hpp
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| template<typename Scalar> | ||
| Scalar const RP2Element<Scalar>::x() const { | ||
| if (x_ == Scalar() && z_ == 0.0) { | ||
| return Infinity<Scalar>(); |
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This infinity should have the sign of x_, since z_ is always positive.
geometry/rp2_element_body.hpp
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| template<typename Scalar> | ||
| Scalar const RP2Element<Scalar>::y() const { | ||
| if (y_ == Scalar() && z_ == 0.0) { | ||
| return Infinity<Scalar>(); |
geometry/rp2_element_body.hpp
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| Scalar const& x, Scalar const& y, double z) | ||
| : x_(x), y_(y), z_(z) { | ||
| // [0:0:0] does not represent any point. | ||
| CHECK(x != Scalar() || y != Scalar() || z != 0.0); |
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I am skeptical; [underflow:underflow:underflow] should probably not cause an outright failure...
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As discussed, removing that check will do the job.
geometry/rp2_element.hpp
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| bool is_at_infinity() const; | ||
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| // Returns the Euclidean (inhomogeneous) coordinates of the point. May be | ||
| // infinities. |
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Document the semantics: the signs of (x,y) will be in the quadrant determined by the signs of x and y, and both will be infinite or none.
geometry/rp2_element.hpp
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| public: | ||
| // Constructs a point from a set of homogeneous coordinates. The | ||
| // coordinates may not all be zero. | ||
| explicit RP2Element(Scalar const& x, Scalar const& y, double z); |
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Document the semantics: this class handles underflows gracefully, and behaves as if taking the limit z->0 before the limit of x and y.
| os << DebugString(rp2_element); | ||
| return os; | ||
| } | ||
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Add some tests with signed 0s exercising the aforementioned behaviour.
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