Line data Source code
1 0 : // Distributed under the MIT License.
2 : // See LICENSE.txt for details.
3 :
4 : #pragma once
5 :
6 : #include <array>
7 : #include <cstddef>
8 : #include <limits>
9 : #include <optional>
10 :
11 : #include "DataStructures/Tensor/TypeAliases.hpp"
12 : #include "Utilities/Gsl.hpp"
13 : #include "Utilities/TypeTraits/RemoveReferenceWrapper.hpp"
14 :
15 : /// \cond
16 : namespace PUP {
17 : class er;
18 : } // namespace PUP
19 : /// \endcond
20 :
21 : /// Contains FocallyLiftedInnerMaps
22 1 : namespace domain::CoordinateMaps::FocallyLiftedInnerMaps {
23 : /*!
24 : * \brief A FocallyLiftedInnerMap that maps a 3D unit right cylinder
25 : * to a volume that connects portions of two spherical surfaces.
26 : *
27 : * Because FocallyLiftedEndcap is a FocallyLiftedInnerMap, it is meant
28 : * to be a template parameter of FocallyLiftedMap, and its member functions
29 : * are meant to be used by FocallyLiftedMap. See FocallyLiftedMap for further
30 : * documentation.
31 : *
32 : * \image html FocallyLiftedEndcap.svg "Focally Lifted Endcap."
33 : *
34 : * \details The domain of the map is a 3D unit right cylinder with
35 : * coordinates \f$(\bar{x},\bar{y},\bar{z})\f$ such that
36 : * \f$-1\leq\bar{z}\leq 1\f$ and \f$\bar{x}^2+\bar{y}^2 \leq
37 : * 1\f$. The range of the map has coordinates \f$(x,y,z)\f$.
38 : *
39 : * Consider a sphere with center \f$C^i\f$ and radius \f$R\f$ that is
40 : * intersected by a plane normal to the \f$z\f$ axis and located at
41 : * \f$z = z_\mathrm{P}\f$. In the figure above, every point
42 : * \f$\bar{x}^i\f$ in the blue region \f$\sigma=0\f$ maps to a point
43 : * \f$x_0^i\f$ on a portion of the surface of the sphere.
44 : *
45 : * `Endcap` provides the following functions:
46 : *
47 : * ### forward_map
48 : * `forward_map` maps \f$(\bar{x},\bar{y},\bar{z}=-1)\f$ to the portion of
49 : * the sphere with \f$z \geq z_\mathrm{P}\f$. The arguments to `forward_map`
50 : * are \f$(\bar{x},\bar{y},\bar{z})\f$, but \f$\bar{z}\f$ is ignored.
51 : * `forward_map` returns \f$x_0^i\f$,
52 : * the 3D coordinates on that sphere, which are given by
53 : *
54 : * \f{align}
55 : * x_0^0 &= R \frac{\sin(\bar{\rho} \theta_\mathrm{max})
56 : * \bar{x}}{\bar{\rho}} + C^0,\\
57 : * x_0^1 &= R \frac{\sin(\bar{\rho} \theta_\mathrm{max})
58 : * \bar{y}}{\bar{\rho}} + C^1,\\
59 : * x_0^2 &= R \cos(\bar{\rho} \theta_\mathrm{max}) + C^2.
60 : * \f}
61 : *
62 : * Here \f$\bar{\rho}^2 \equiv (\bar{x}^2+\bar{y}^2)/\bar{R}^2\f$, where
63 : * \f$\bar{R}\f$ is the radius of the cylinder in barred coordinates,
64 : * which is always unity,
65 : * and where
66 : * \f$\theta_\mathrm{max}\f$ is defined by
67 : * \f$\cos(\theta_\mathrm{max}) = (z_\mathrm{P}-C^2)/R\f$.
68 : * Note that when \f$\bar{\rho}=0\f$, we must evaluate
69 : * \f$\sin(\bar{\rho}\theta_\mathrm{max})/\bar{\rho}\f$
70 : * as \f$\theta_\mathrm{max}\f$.
71 : *
72 : * ### sigma
73 : *
74 : * \f$\sigma\f$ is a function that is zero at \f$\bar{z}=-1\f$
75 : * (which maps onto the sphere \f$x^i=x_0^i\f$) and
76 : * unity at \f$\bar{z}=+1\f$ (corresponding to the
77 : * upper surface of the FocallyLiftedMap). We define
78 : *
79 : * \f{align}
80 : * \sigma &= \frac{\bar{z}+1}{2}.
81 : * \f}
82 : *
83 : * ### deriv_sigma
84 : *
85 : * `deriv_sigma` returns
86 : *
87 : * \f{align}
88 : * \frac{\partial \sigma}{\partial \bar{x}^j} &= (0,0,1/2).
89 : * \f}
90 : *
91 : * ### jacobian
92 : *
93 : * `jacobian` returns \f$\partial x_0^k/\partial \bar{x}^j\f$.
94 : * The arguments to `jacobian`
95 : * are \f$(\bar{x},\bar{y},\bar{z})\f$, but \f$\bar{z}\f$ is ignored.
96 : *
97 : * Differentiating Eqs.(1--3) above yields
98 : *
99 : * \f{align*}
100 : * \frac{\partial x_0^2}{\partial \bar{x}} &=
101 : * - R \theta_\mathrm{max}
102 : * \frac{\sin(\bar{\rho}\theta_\mathrm{max})}{\bar{\rho}}\bar{x}\\
103 : * \frac{\partial x_0^2}{\partial \bar{y}} &=
104 : * - R \theta_\mathrm{max}
105 : * \frac{\sin(\bar{\rho}\theta_\mathrm{max})}{\bar{\rho}}\bar{y}\\
106 : * \frac{\partial x_0^0}{\partial \bar{x}} &=
107 : * R \frac{\sin(\bar{\rho}\theta_\mathrm{max})}{\bar{\rho}} +
108 : * R \frac{1}{\bar{\rho}}\frac{d}{d\bar{\rho}}
109 : * \left(\frac{\sin(\bar{\rho}\theta_\mathrm{max})}{\bar{\rho}}\right)
110 : * \bar{x}^2\\
111 : * \frac{\partial x_0^0}{\partial \bar{y}} &=
112 : * R \frac{1}{\bar{\rho}}\frac{d}{d\bar{\rho}}
113 : * \left(\frac{\sin(\bar{\rho}\theta_\mathrm{max})}{\bar{\rho}}\right)
114 : * \bar{x}\bar{y}\\
115 : * \frac{\partial x_0^1}{\partial \bar{x}} &=
116 : * R \frac{1}{\bar{\rho}}\frac{d}{d\bar{\rho}}
117 : * \left(\frac{\sin(\bar{\rho}\theta_\mathrm{max})}{\bar{\rho}}\right)
118 : * \bar{x}\bar{y}\\
119 : * \frac{\partial x_0^1}{\partial \bar{y}} &=
120 : * R \frac{\sin(\bar{\rho}\theta_\mathrm{max})}{\bar{\rho}} +
121 : * R \frac{1}{\bar{\rho}}\frac{d}{d\bar{\rho}}
122 : * \left(\frac{\sin(\bar{\rho}\theta_\mathrm{max})}{\bar{\rho}}\right)
123 : * \bar{y}^2\\
124 : * \frac{\partial x_0^i}{\partial \bar{z}} &=0.
125 : * \f}
126 : *
127 : * ### Evaluating sinc functions
128 : *
129 : * Note that \f$\sin(\bar{\rho}\theta_\mathrm{max})/\bar{\rho}\f$ and
130 : * its derivative appear in the above equations. We evaluate
131 : * \f$\sin(ax)/x\f$ in a straightforward way, except we are careful
132 : * to evaluate \f$\sin(ax)/x = a\f$ for the special case \f$x=0\f$.
133 : *
134 : * The derivative of the sync function is more complicated to evaluate
135 : * because of roundoff. Note that we can expand
136 : *
137 : * \f{align*}
138 : * \frac{1}{x}\frac{d}{dx}\left(\frac{\sin(ax)}{x}\right) &=
139 : * \frac{a}{x^2}\left(1 - 1 - \frac{2 (ax)^2}{3!} +
140 : * \frac{4(ax)^4}{5!} - \frac{5(ax)^6}{7!} + \cdots \right),
141 : * \f}
142 : *
143 : * where we kept the "1 - 1" above as a reminder that when evaluating
144 : * this function directly as \f$(a \cos(ax)/x^2 - \sin(ax)/x^3)\f$, there
145 : * can be significant roundoff because of the "1" in each of the two
146 : * terms that are subtracted. The relative error in direct evaluation is
147 : * \f$3\epsilon/(ax)^2\f$, where \f$\epsilon\f$ is machine precision
148 : * (This expression comes from replacing "1 - 1" with \f$\epsilon\f$
149 : * and comparing the lowest-order contribution to the correct answer, i.e.
150 : * the \f$2(ax)^2/3!\f$ term, with \f$\epsilon\f$, the error contribution).
151 : * This means the error is 100% if \f$(ax)^2=3\epsilon\f$.
152 : *
153 : * To avoid roundoff, we evaluate the series if \f$ax\f$ is small
154 : * enough. Suppose we keep terms up to and including the \f$(ax)^{2n}\f$
155 : * term in the series. Then we evaluate the series if the
156 : * next term, the \f$(ax)^{2n+2}\f$ term, is roundoff,
157 : * i.e. if \f$(2n+2)(ax)^{2n+2}/(2n+3)! < \epsilon\f$. In this case,
158 : * the direct evaluation has the maximum error if
159 : * \f$(2n+2)(ax)^{2n+2}/(2n+3)! = \epsilon\f$. We showed above that the
160 : * relative error in direct evaluation is \f$3\epsilon/(ax)^2\f$,
161 : * which evaluates to \f$(\epsilon^n (2n+2)/(2n+3)!)^{1/(n+1)}\f$.
162 : *
163 : * \f{align*}
164 : * n=1 \qquad& \mathrm{error}=3(\epsilon/30)^{1/2} &\qquad
165 : * \sim \mathrm{5e-09}\\
166 : * n=2 \qquad& \mathrm{error}=3(\epsilon^2/840)^{1/3} &\qquad
167 : * \sim \mathrm{7e-12}\\
168 : * n=3 \qquad& \mathrm{error}=3(\epsilon^3/45360)^{1/4} &\qquad
169 : * \sim \mathrm{2e-13}\\
170 : * n=4 \qquad& \mathrm{error}=3(\epsilon^4/3991680)^{1/5} &\qquad
171 : * \sim \mathrm{2e-14}\\
172 : * n=5 \qquad& \mathrm{error}=3(\epsilon^5/518918400)^{1/6} &\qquad
173 : * \sim \mathrm{5e-15}\\
174 : * n=6 \qquad& \mathrm{error}=3(\epsilon^6/93405312000)^{1/7} &\qquad
175 : * \sim \mathrm{1e-15}
176 : * \f}
177 : * We gain less and less with each order, so we choose \f$n=3\f$.
178 : * Then the series above can be rewritten to this order in the form
179 : * \f{align*}
180 : * \frac{1}{x}\frac{d}{dx}\left(\frac{\sin(ax)}{x}\right) &=
181 : * -\frac{a^3}{3}\left(1 - \frac{3\cdot 4 (ax)^2}{5!} +
182 : * \frac{3 \cdot 6(ax)^4}{7!}\right).
183 : * \f}
184 : *
185 : * ### inverse
186 : *
187 : * `inverse` takes \f$x_0^i\f$ and \f$\sigma\f$ as arguments, and
188 : * returns \f$(\bar{x},\bar{y},\bar{z})\f$, or boost::none if
189 : * \f$x_0^i\f$ or \f$\sigma\f$ are outside the range of the map.
190 : * For example, if \f$x_0^i\f$ does not lie on the sphere,
191 : * we return boost::none.
192 : *
193 : * The easiest to compute is \f$\bar{z}\f$, which is given by inverting
194 : * Eq. (4):
195 : *
196 : * \f{align}
197 : * \bar{z} &= 2\sigma - 1.
198 : * \f}
199 : *
200 : * If \f$\bar{z}\f$ is outside the range \f$[-1,1]\f$ then we return
201 : * boost::none.
202 : *
203 : * To get \f$\bar{x}\f$ and \f$\bar{y}\f$,
204 : * we invert
205 : * Eqs (1--3). If \f$x_0^0=x_0^1=0\f$, then \f$\bar{x}=\bar{y}=0\f$.
206 : * Otherwise, we compute
207 : *
208 : * \f{align}
209 : * \bar{\rho} = \theta_\mathrm{max}^{-1}
210 : * \tan^{-1}\left(\frac{\rho}{x_0^2-C^2}\right),
211 : * \f}
212 : *
213 : * where \f$\rho^2 = (x_0^0-C^0)^2+(x_0^1-C^1)^2\f$. Then
214 : *
215 : * \f{align}
216 : * \bar{x} &= (x_0^0-C^0)\frac{\bar{\rho}}{\rho},\\
217 : * \bar{y} &= (x_0^1-C^1)\frac{\bar{\rho}}{\rho}.
218 : * \f}
219 : *
220 : * Note that if \f$\bar{x}^2+\bar{y}^2 > 1\f$, the original point is outside
221 : * the range of the map so we return boost::none.
222 : *
223 : * ### lambda_tilde
224 : *
225 : * `lambda_tilde` takes as arguments a point \f$x^i\f$ and a projection point
226 : * \f$P^i\f$, and computes \f$\tilde{\lambda}\f$, the solution to
227 : *
228 : * \f{align} x_0^i = P^i + (x^i - P^i) \tilde{\lambda}.\f}
229 : *
230 : * Since \f$x_0^i\f$ must lie on the sphere, \f$\tilde{\lambda}\f$ is the
231 : * solution of the quadratic equation
232 : *
233 : * \f{align}
234 : * |P^i + (x^i - P^i) \tilde{\lambda} - C^i |^2 - R^2 = 0.
235 : * \f}
236 : *
237 : * In solving the quadratic, we demand a root that is positive and
238 : * less than or equal to unity, since \f$x_0^i\f$ is always between
239 : * the projection point and \f$x^i\f$. If there are two suitable
240 : * roots, this means that the entire sphere lies between \f$P^i\f$ and
241 : * \f$x^i\f$; in this case if \f$x^2 \geq z_\mathrm{P}\f$ we choose the
242 : * larger root, otherwise we choose the smaller one: this gives
243 : * us the root with \f$x_0^2 \geq z_\mathrm{P}\f$, the portion of the sphere
244 : * that is the range of `Endcap`. If there is no suitable root,
245 : * this means that the point \f$x^i\f$ is not in the range of the map
246 : * so we return a default-constructed std::optional.
247 : *
248 : * ### deriv_lambda_tilde
249 : *
250 : * `deriv_lambda_tilde` takes as arguments \f$x_0^i\f$, a projection point
251 : * \f$P^i\f$, and \f$\tilde{\lambda}\f$, and
252 : * returns \f$\partial \tilde{\lambda}/\partial x^i\f$.
253 : * By differentiating Eq. (11), we find
254 : *
255 : * \f{align}
256 : * \frac{\partial\tilde{\lambda}}{\partial x^j} &=
257 : * \tilde{\lambda}^2 \frac{C^j - x_0^j}{|x_0^i - P^i|^2
258 : * + (x_0^i - P^i)(P_i - C_{i})}.
259 : * \f}
260 : *
261 : * ### inv_jacobian
262 : *
263 : * `inv_jacobian` returns \f$\partial \bar{x}^i/\partial x_0^k\f$,
264 : * where \f$\sigma\f$ is held fixed.
265 : * The arguments to `inv_jacobian`
266 : * are \f$(\bar{x},\bar{y},\bar{z})\f$, but \f$\bar{z}\f$ is ignored.
267 : *
268 : * Note that \f$\bar{x}\f$ and \f$\bar{y}\f$ can be considered to
269 : * depend only on \f$x_0^0\f$ and \f$x_0^1\f$ but not on \f$x_0^2\f$,
270 : * because the point \f$x_0^i\f$ is constrained to lie on a sphere of
271 : * radius \f$R\f$. Note that there is an alternative way to compute
272 : * Eqs. (8) and (9) using only \f$x_0^0\f$ and \f$x_0^1\f$. To do
273 : * this, define
274 : *
275 : * \f{align}
276 : * \upsilon \equiv \sin(\bar{\rho}\theta_\mathrm{max})
277 : * &= \sqrt{\frac{(x_0^0-C^0)^2+(x_0^1-C^1)^2}{R^2}}.
278 : * \f}
279 : *
280 : * Then we can write
281 : *
282 : * \f{align}
283 : * \frac{1}{\bar{\rho}}\sin(\bar{\rho}\theta_\mathrm{max})
284 : * &= \frac{\theta_\mathrm{max}\upsilon}{\arcsin(\upsilon)},
285 : * \f}
286 : *
287 : * so that
288 : *
289 : * \f{align}
290 : * \bar{x} &= \frac{x_0^0-C^0}{R}\left(\frac{1}{\bar{\rho}}
291 : * \sin(\bar{\rho}\theta_\mathrm{max})\right)^{-1} \\
292 : * \bar{y} &= \frac{x_0^1-C^1}{R}\left(\frac{1}{\bar{\rho}}
293 : * \sin(\bar{\rho}\theta_\mathrm{max})\right)^{-1}.
294 : * \f}
295 : *
296 : * We will compute \f$\partial \bar{x}^i/\partial
297 : * x_0^k\f$ by differentiating Eqs. (15) and (16). Because those equations
298 : * involve \f$\bar{\rho}\f$, we first establish some relations
299 : * involving derivatives of \f$\bar{\rho}\f$. For ease of notation, we define
300 : *
301 : * \f{align}
302 : * q \equiv \frac{\sin(\bar{\rho}\theta_\mathrm{max})}{\bar{\rho}}.
303 : * \f}
304 : *
305 : * First observe that
306 : * \f{align}
307 : * \frac{dq}{d\upsilon}
308 : * = \frac{dq}{d\bar{\rho}}
309 : * \left(\bar{\rho} \frac{dq}{d\bar{\rho}} + q\right)^{-1},
310 : * \f}
311 : *
312 : * where \f$\upsilon\f$ is the quantity defined by Eq. (13). Therefore
313 : *
314 : * \f{align}
315 : * \frac{\partial q}{\partial x_0^0} &=
316 : * \frac{\bar{x}}{\bar{\rho}R}\frac{dq}{d\bar{\rho}}
317 : * \left(\bar{\rho} \frac{dq}{d\bar{\rho}} + q\right)^{-1},\\
318 : * \frac{\partial q}{\partial x_0^1} &=
319 : * \frac{\bar{y}}{\bar{\rho}R}\frac{dq}{d\bar{\rho}}
320 : * \left(\bar{\rho} \frac{dq}{d\bar{\rho}} + q\right)^{-1},
321 : * \f}
322 : *
323 : * where we have differentiated Eq. (13), and where we have
324 : * used Eqs. (15) and (16) to eliminate \f$x_0^0\f$ and
325 : * \f$x_0^1\f$ in favor of \f$\bar{x}\f$ and
326 : * \f$\bar{y}\f$ in the final result.
327 : *
328 : * Let
329 : * \f{align}
330 : * \Sigma \equiv \frac{1}{\bar\rho} \frac{dq}{d\bar{\rho}},
331 : * \f}
332 : * since that combination will appear frequently in formulas below. Note that
333 : * \f$\Sigma\f$ has a finite limit as \f$\bar{\rho}\to 0\f$, and it is evaluated
334 : * according to the section on evaluating sinc functions above.
335 : *
336 : * By differentiating Eqs. (15) and (16), and using Eqs. (19) and (20), we
337 : * find
338 : *
339 : * \f{align}
340 : * \frac{\partial \bar{x}}{\partial x_0^0} &=
341 : * \frac{1}{R q}
342 : * - \frac{\bar{x}^2 \Sigma}{R q}
343 : * \left(\bar{\rho}^2 \Sigma + q\right)^{-1},\\
344 : * \frac{\partial \bar{x}}{\partial x_0^1} &=
345 : * - \frac{\bar{x}\bar{y} \Sigma}{R q}
346 : * \left(\bar{\rho}^2 \Sigma + q\right)^{-1},\\
347 : * \frac{\partial \bar{x}}{\partial x_0^2} &= 0,\\
348 : * \frac{\partial \bar{y}}{\partial x_0^0} &=
349 : * \frac{\partial \bar{x}}{\partial x_0^1},\\
350 : * \frac{\partial \bar{y}}{\partial x_0^1} &=
351 : * \frac{1}{R q}
352 : * - \frac{\bar{y}^2 \Sigma}{R q}
353 : * \left(\bar{\rho}^2 \Sigma + q\right)^{-1},\\
354 : * \frac{\partial \bar{y}}{\partial x_0^2} &= 0,\\
355 : * \frac{\partial \bar{z}}{\partial x_0^i} &= 0.
356 : * \f}
357 : * Note that care must be taken to evaluate
358 : * \f$q = \sin(\bar{\rho}\theta_\mathrm{max})/\bar{\rho}\f$ and its
359 : * derivative \f$\Sigma\f$ near \f$\bar{\rho}=0\f$; see the discussion above on
360 : * evaluating sinc functions.
361 : *
362 : * ### dxbar_dsigma
363 : *
364 : * `dxbar_dsigma` returns \f$\partial \bar{x}^i/\partial \sigma\f$,
365 : * where \f$x_0^i\f$ is held fixed.
366 : *
367 : * From Eq. (6) we have
368 : *
369 : * \f{align}
370 : * \frac{\partial \bar{x}^i}{\partial \sigma} &= (0,0,2).
371 : * \f}
372 : *
373 : */
374 1 : class Endcap {
375 : public:
376 0 : Endcap(const std::array<double, 3>& center, double radius, double z_plane);
377 :
378 0 : Endcap() = default;
379 0 : ~Endcap() = default;
380 0 : Endcap(Endcap&&) = default;
381 0 : Endcap(const Endcap&) = default;
382 0 : Endcap& operator=(const Endcap&) = default;
383 0 : Endcap& operator=(Endcap&&) = default;
384 :
385 : template <typename T>
386 0 : void forward_map(
387 : const gsl::not_null<std::array<tt::remove_cvref_wrap_t<T>, 3>*>
388 : target_coords,
389 : const std::array<T, 3>& source_coords) const;
390 :
391 : /// The inverse function is only callable with doubles because the inverse
392 : /// might fail if called for a point out of range, and it is unclear
393 : /// what should happen if the inverse were to succeed for some points in a
394 : /// DataVector but fail for other points.
395 1 : std::optional<std::array<double, 3>> inverse(
396 : const std::array<double, 3>& target_coords, double sigma_in) const;
397 :
398 : template <typename T>
399 0 : void jacobian(const gsl::not_null<
400 : tnsr::Ij<tt::remove_cvref_wrap_t<T>, 3, Frame::NoFrame>*>
401 : jacobian_out,
402 : const std::array<T, 3>& source_coords) const;
403 :
404 : template <typename T>
405 0 : void inv_jacobian(const gsl::not_null<tnsr::Ij<tt::remove_cvref_wrap_t<T>, 3,
406 : Frame::NoFrame>*>
407 : inv_jacobian_out,
408 : const std::array<T, 3>& source_coords) const;
409 :
410 : template <typename T>
411 0 : void sigma(const gsl::not_null<tt::remove_cvref_wrap_t<T>*> sigma_out,
412 : const std::array<T, 3>& source_coords) const;
413 :
414 : template <typename T>
415 0 : void deriv_sigma(
416 : const gsl::not_null<std::array<tt::remove_cvref_wrap_t<T>, 3>*>
417 : deriv_sigma_out,
418 : const std::array<T, 3>& source_coords) const;
419 :
420 : template <typename T>
421 0 : void dxbar_dsigma(
422 : const gsl::not_null<std::array<tt::remove_cvref_wrap_t<T>, 3>*>
423 : dxbar_dsigma_out,
424 : const std::array<T, 3>& source_coords) const;
425 :
426 0 : std::optional<double> lambda_tilde(
427 : const std::array<double, 3>& parent_mapped_target_coords,
428 : const std::array<double, 3>& projection_point,
429 : bool source_is_between_focus_and_target) const;
430 :
431 : template <typename T>
432 0 : void deriv_lambda_tilde(
433 : const gsl::not_null<std::array<tt::remove_cvref_wrap_t<T>, 3>*>
434 : deriv_lambda_tilde_out,
435 : const std::array<T, 3>& target_coords, const T& lambda_tilde,
436 : const std::array<double, 3>& projection_point) const;
437 :
438 : // NOLINTNEXTLINE(google-runtime-references)
439 0 : void pup(PUP::er& p);
440 :
441 0 : static bool is_identity() { return false; }
442 :
443 : private:
444 0 : friend bool operator==(const Endcap& lhs, const Endcap& rhs);
445 0 : std::array<double, 3> center_{};
446 0 : double radius_{std::numeric_limits<double>::signaling_NaN()};
447 0 : double theta_max_{std::numeric_limits<double>::signaling_NaN()};
448 : };
449 0 : bool operator!=(const Endcap& lhs, const Endcap& rhs);
450 : } // namespace domain::CoordinateMaps::FocallyLiftedInnerMaps
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