% Input: frame declarations and placement requests from front ends; the metric registry. % Output: resolved coordinates, world angles, and silhouette support distances. % Owned state: the frame table, the placement request store, and resolved positions. % Invariants: positions resolve topologically or error; nothing here emits ink. % Next stage: rendering turns resolved records into marks on the page. % SPDX-License-Identifier: Apache-2.0 % Copyright the TNLean project; see LICENSE for the full terms. % % Geometry answers three questions and nothing else. WHERE: frames map % logical addresses to points (one 2x2 matrix plus offset, or a circle of % stations), and a dependency-graph evaluator resolves relative placements % in one topological pass -- a cycle is a coded error naming the cycle. % WHICH WAY: a direction crosses a frame through its linear part, so a % rotation is angle addition and a compass word is converted once. % HOW FAR: a silhouette is a support function -- the reach of a rotated % skin along a query direction -- and clearance questions fold max-support % over record sets. All lengths here are pitch factors; only rendering % multiplies by the pitch dimension. \ExplSyntaxOn % ---------- messages ------------------------------------------------------- \msg_new:nnn {tenkz}{geom-unknown-frame} { [TKZ-GEOM-UNKNOWN-FRAME]~no~frame~named~'#1' } \msg_new:nnn {tenkz}{geom-frame-kind} { [TKZ-GEOM-FRAME-KIND]~frame~'#1'~is~a~'#2'~frame;~#3~needs~a~'#4'~frame } \msg_new:nnn {tenkz}{geom-compose} { [TKZ-GEOM-COMPOSE]~only~affine~frames~compose;~'#1'~is~not~affine } \msg_new:nnn {tenkz}{geom-unknown-place} { [TKZ-GEOM-UNPLACED]~no~placement~named~'#1' } \msg_new:nnn {tenkz}{geom-place-cycle} { [TKZ-PLACE-CYCLE]~placements~depend~on~themselves:~#1 } \msg_new:nnn {tenkz}{geom-family} { [TKZ-GEOM-FAMILY]~no~silhouette~family~named~'#1' } \msg_new:nnn {tenkz}{geom-compass} { [TKZ-GEOM-COMPASS]~no~compass~face~named~'#1';~expected~n,~e,~s,~or~w } \msg_new:nnn {tenkz}{geom-singular-basis} { [TKZ-GEOM-SINGULAR-BASIS]~a~carrier~basis~must~have~nonzero~determinant } % ---------- frames ---------------------------------------------------------- % A frame record is six fp fields (row-major matrix a b / c d, offset dx dy) % for kind=affine, or fields n (stations), r (radius), start (degrees) for % kind=circle. A projected plane additionally owns one page-space transverse % vector. Frames are picture-local. \__tenkz_prop_new_indexed:N \l__tenkz_geom_frame_prop % "name/field" -> fp literal \prop_new:N \l__tenkz_geom_framekind_prop % "name" -> affine | circle \fp_const:Nn \c__tenkz_geom_plane_transverse_x_fp {0} \fp_const:Nn \c__tenkz_geom_plane_transverse_y_fp {1} \cs_new_protected:Npn \__tenkz_geom_frame_put:nnn #1#2#3 { \prop_put:Nen \l__tenkz_geom_frame_prop { #1 / #2 } { \fp_eval:n {#3} } } \cs_new_protected:Npn \__tenkz_geom_frame_declare_transverse:nnn #1#2#3 { \__tenkz_geom_frame_put:nnn {#1} {transverse-x} {#2} \__tenkz_geom_frame_put:nnn {#1} {transverse-y} {#3} } \cs_new_protected:Npn \__tenkz_geom_frame_declare_affine:nnnnnnn #1#2#3#4#5#6#7 { \prop_put:Nnn \l__tenkz_geom_framekind_prop {#1} {affine} \__tenkz_geom_frame_put:nnn {#1} {a} {#2} \__tenkz_geom_frame_put:nnn {#1} {b} {#3} \__tenkz_geom_frame_put:nnn {#1} {c} {#4} \__tenkz_geom_frame_put:nnn {#1} {d} {#5} \__tenkz_geom_frame_put:nnn {#1} {dx} {#6} \__tenkz_geom_frame_put:nnn {#1} {dy} {#7} } % The public plane word has one fixed projected basis. Logical coordinates % arrive as (row, column), with (1,1) at the origin: C=(1,0) and % R=(-planeslant,-planerise). Keeping this declaration beside the other % frame maps gives every front end the same projection and leaves rendering % responsible only for ink. \cs_new_protected:Npn \__tenkz_geom_frame_declare_plane:n #1 { \__tenkz_geom_frame_declare_affine:nnnnnnn {#1} { -\__tenkz_metric_ratio:n {planeslant} } {1} { -\__tenkz_metric_ratio:n {planerise} } {0} { \__tenkz_metric_ratio:n {planeslant} - 1 } { \__tenkz_metric_ratio:n {planerise} } % The projected carrier is the zx plane; its physical y axis stays % vertical on the page and is not a direction in the affine plane basis. \__tenkz_geom_frame_declare_transverse:nnn {#1} { \c__tenkz_geom_plane_transverse_x_fp } { \c__tenkz_geom_plane_transverse_y_fp } } % flat is the identity; a rotation frame is one call with cosd/sind \cs_new_protected:Npn \__tenkz_geom_frame_declare_rotation:nnnn #1#2#3#4 { \__tenkz_geom_frame_declare_affine:nnnnnnn {#1} { cosd(#2) } { -sind(#2) } { sind(#2) } { cosd(#2) } {#3} {#4} } \cs_new_protected:Npn \__tenkz_geom_frame_declare_circle:nnnn #1#2#3#4 { \prop_put:Nnn \l__tenkz_geom_framekind_prop {#1} {circle} \__tenkz_geom_frame_put:nnn {#1} {n} {#2} \__tenkz_geom_frame_put:nnn {#1} {r} {#3} \__tenkz_geom_frame_put:nnn {#1} {start} {#4} } \cs_new:Npn \__tenkz_geom_field:nn #1#2 { \prop_item:Nn \l__tenkz_geom_frame_prop { #1 / #2 } } % Return the signed page vector on the independent transverse axis. Only a % frame which declares that third carrier direction may call this service; % flat and circular physical directions continue through their local bases. \cs_new_protected:Npn \__tenkz_geom_frame_transverse:nnNN #1#2#3#4 { \str_if_eq:nnTF {#2} {n} { \tl_set:Ne #3 { \__tenkz_geom_field:nn {#1} {transverse-x} } \tl_set:Ne #4 { \__tenkz_geom_field:nn {#1} {transverse-y} } } { \tl_set:Ne #3 { -\__tenkz_geom_field:nn {#1} {transverse-x} } \tl_set:Ne #4 { -\__tenkz_geom_field:nn {#1} {transverse-y} } } } \cs_new_protected:Npn \__tenkz_geom_require_kind:nnnN #1#2#3#4 { \bool_set_false:N #4 \prop_get:NnN \l__tenkz_geom_framekind_prop {#1} \l__tenkz_geom_tmp_tl \quark_if_no_value:NTF \l__tenkz_geom_tmp_tl { \msg_error:nnn {tenkz}{geom-unknown-frame} {#1} } { \str_if_eq:VnTF \l__tenkz_geom_tmp_tl {#2} { \bool_set_true:N #4 } { \msg_error:nnxxxx {tenkz}{geom-frame-kind} {#1} { \l__tenkz_geom_tmp_tl } {#3} {#2} } } } \tl_new:N \l__tenkz_geom_tmp_tl \bool_new:N \l__tenkz_geom_kind_ok_bool % new = outer applied after inner: M = Mo Mi, t = Mo ti + to \cs_new_protected:Npn \__tenkz_geom_frame_compose:nnn #1#2#3 { \__tenkz_geom_require_kind:nnnN {#2} {affine} {compose} \l__tenkz_geom_kind_ok_bool \bool_if:NT \l__tenkz_geom_kind_ok_bool { \__tenkz_geom_require_kind:nnnN {#3} {affine} {compose} \l__tenkz_geom_kind_ok_bool } \bool_if:NT \l__tenkz_geom_kind_ok_bool { \__tenkz_geom_frame_declare_affine:nnnnnnn {#1} { \__tenkz_geom_field:nn{#2}{a} * \__tenkz_geom_field:nn{#3}{a} + \__tenkz_geom_field:nn{#2}{b} * \__tenkz_geom_field:nn{#3}{c} } { \__tenkz_geom_field:nn{#2}{a} * \__tenkz_geom_field:nn{#3}{b} + \__tenkz_geom_field:nn{#2}{b} * \__tenkz_geom_field:nn{#3}{d} } { \__tenkz_geom_field:nn{#2}{c} * \__tenkz_geom_field:nn{#3}{a} + \__tenkz_geom_field:nn{#2}{d} * \__tenkz_geom_field:nn{#3}{c} } { \__tenkz_geom_field:nn{#2}{c} * \__tenkz_geom_field:nn{#3}{b} + \__tenkz_geom_field:nn{#2}{d} * \__tenkz_geom_field:nn{#3}{d} } { \__tenkz_geom_field:nn{#2}{a} * \__tenkz_geom_field:nn{#3}{dx} + \__tenkz_geom_field:nn{#2}{b} * \__tenkz_geom_field:nn{#3}{dy} + \__tenkz_geom_field:nn{#2}{dx} } { \__tenkz_geom_field:nn{#2}{c} * \__tenkz_geom_field:nn{#3}{dx} + \__tenkz_geom_field:nn{#2}{d} * \__tenkz_geom_field:nn{#3}{dy} + \__tenkz_geom_field:nn{#2}{dy} } } } % apply a frame to a logical point; results are fp literals in pitch units. % A circle frame reads the second coordinate as a station and the first as a % step inward, so the flat frame's rightward column becomes travel along the % ring and its downward row becomes travel toward the centre. Both readings % accept fractions: a port half a cell east of its atom is half a station % further round. \cs_new_protected:Npn \__tenkz_geom_apply:nnnNNN #1#2#3#4#5#6 { \tl_clear:N #4 \tl_clear:N #5 \bool_set_false:N #6 \prop_get:NnN \l__tenkz_geom_framekind_prop {#1} \l__tenkz_geom_kind_tl \quark_if_no_value:NTF \l__tenkz_geom_kind_tl { \msg_error:nnn {tenkz}{geom-unknown-frame} {#1} } { \bool_set_true:N #6 \str_if_eq:VnTF \l__tenkz_geom_kind_tl {circle} { \tl_set:Ne #4 { \fp_eval:n { ( \__tenkz_geom_field:nn{#1}{r} - ( (#2) - 1 ) ) * cosd( \__tenkz_geom_station_angle:nn {#1} {#3} ) } } \tl_set:Ne #5 { \fp_eval:n { ( \__tenkz_geom_field:nn{#1}{r} - ( (#2) - 1 ) ) * sind( \__tenkz_geom_station_angle:nn {#1} {#3} ) } } } { \tl_set:Ne #4 { \fp_eval:n { \__tenkz_geom_field:nn{#1}{a} * (#2) + \__tenkz_geom_field:nn{#1}{b} * (#3) + \__tenkz_geom_field:nn{#1}{dx} } } \tl_set:Ne #5 { \fp_eval:n { \__tenkz_geom_field:nn{#1}{c} * (#2) + \__tenkz_geom_field:nn{#1}{d} * (#3) + \__tenkz_geom_field:nn{#1}{dy} } } } } } \tl_new:N \l__tenkz_geom_kind_tl \cs_new_protected:Npn \__tenkz_geom_apply:nnnNN #1#2#3#4#5 { \__tenkz_geom_apply:nnnNNN {#1}{#2}{#3} #4 #5 \l__tenkz_geom_kind_ok_bool } % Recover the logical point under an affine frame. Open-end projection uses % this inverse so it can keep the neighbouring point's transverse logical % coordinate before placing the tip on a projected margin; mixing a page x % with a separately projected page y is valid only for an axis-aligned frame. \cs_new_protected:Npn \__tenkz_geom_unapply:nnnNN #1#2#3#4#5 { \tl_clear:N #4 \tl_clear:N #5 \__tenkz_geom_require_kind:nnnN {#1} {affine} {unapply} \l__tenkz_geom_kind_ok_bool \bool_if:NT \l__tenkz_geom_kind_ok_bool { \tl_set:Ne #4 { \fp_eval:n { ( \__tenkz_geom_field:nn{#1}{d} * ( (#2) - \__tenkz_geom_field:nn{#1}{dx} ) - \__tenkz_geom_field:nn{#1}{b} * ( (#3) - \__tenkz_geom_field:nn{#1}{dy} ) ) / ( \__tenkz_geom_field:nn{#1}{a} * \__tenkz_geom_field:nn{#1}{d} - \__tenkz_geom_field:nn{#1}{b} * \__tenkz_geom_field:nn{#1}{c} ) } } \tl_set:Ne #5 { \fp_eval:n { ( -\__tenkz_geom_field:nn{#1}{c} * ( (#2) - \__tenkz_geom_field:nn{#1}{dx} ) + \__tenkz_geom_field:nn{#1}{a} * ( (#3) - \__tenkz_geom_field:nn{#1}{dy} ) ) / ( \__tenkz_geom_field:nn{#1}{a} * \__tenkz_geom_field:nn{#1}{d} - \__tenkz_geom_field:nn{#1}{b} * \__tenkz_geom_field:nn{#1}{c} ) } } } } \prg_new_conditional:Npnn \__tenkz_geom_frame_if_affine:n #1 { T, F, TF } { \prop_get:NnN \l__tenkz_geom_framekind_prop {#1} \l__tenkz_geom_kind_tl \str_if_eq:VnTF \l__tenkz_geom_kind_tl {affine} { \prg_return_true: } { \prg_return_false: } } % a direction crosses the linear part; shear and scale bend it honestly \cs_new_protected:Npn \__tenkz_geom_dir:nnN #1#2#3 { \tl_clear:N #3 \__tenkz_geom_require_kind:nnnN {#1} {affine} {dir} \l__tenkz_geom_kind_ok_bool \bool_if:NT \l__tenkz_geom_kind_ok_bool { \tl_set:Ne #3 { \fp_eval:n { atand ( \__tenkz_geom_field:nn{#1}{c} * cosd(#2) + \__tenkz_geom_field:nn{#1}{d} * sind(#2) , \__tenkz_geom_field:nn{#1}{a} * cosd(#2) + \__tenkz_geom_field:nn{#1}{b} * sind(#2) ) } } } } % A carrier basis is the complete local east/north pair on the page. Keeping % both vectors, rather than one turn angle, preserves shear and unequal scale. % The generic apply/unapply pair below is shared by every consumer which % stores geometry in carrier-local coordinates. \cs_new_protected:Npn \__tenkz_geom_basis_apply:nnnnnnNN #1#2#3#4#5#6#7#8 { \tl_set:Ne #7 { \fp_eval:n { (#1) * (#5) + (#3) * (#6) } } \tl_set:Ne #8 { \fp_eval:n { (#2) * (#5) + (#4) * (#6) } } } % Carry a local direction through a complete carrier basis and read its page % bearing. A scalar turn cannot answer this question under shear or unequal % axis scale; the two basis vectors are the one source of direction geometry. \tl_new:N \l__tenkz_geom_bearing_x_tl \tl_new:N \l__tenkz_geom_bearing_y_tl \cs_new_protected:Npn \__tenkz_geom_basis_bearing:nnnnnN #1#2#3#4#5#6 { \__tenkz_geom_basis_apply:nnnnnnNN {#1}{#2}{#3}{#4} { cosd(#5) } { sind(#5) } \l__tenkz_geom_bearing_x_tl \l__tenkz_geom_bearing_y_tl \tl_set:Ne #6 { \fp_eval:n { atand( \l__tenkz_geom_bearing_y_tl , \l__tenkz_geom_bearing_x_tl ) } } } \cs_new_protected:Npn \__tenkz_geom_basis_unapply:nnnnnnNN #1#2#3#4#5#6#7#8 { \fp_compare:nNnTF { (#1) * (#4) - (#2) * (#3) } = {0} { \tl_clear:N #7 \tl_clear:N #8 \msg_error:nn {tenkz}{geom-singular-basis} } { \tl_set:Ne #7 { \fp_eval:n { ( (#4) * (#5) - (#3) * (#6) ) / ( (#1) * (#4) - (#2) * (#3) ) } } \tl_set:Ne #8 { \fp_eval:n { ( -(#2) * (#5) + (#1) * (#6) ) / ( (#1) * (#4) - (#2) * (#3) ) } } } } % The dual basis reads page vectors as local coordinates. These covectors % also let support folds measure a rendered silhouette directly in local % coordinates without first replacing it by an axis-aligned page box. \cs_new_protected:Npn \__tenkz_geom_basis_dual:nnnnNNNN #1#2#3#4#5#6#7#8 { \fp_compare:nNnTF { (#1) * (#4) - (#2) * (#3) } = {0} { \tl_clear:N #5 \tl_clear:N #6 \tl_clear:N #7 \tl_clear:N #8 \msg_error:nn {tenkz}{geom-singular-basis} } { \tl_set:Ne #5 { \fp_eval:n { (#4) / ( (#1) * (#4) - (#2) * (#3) ) } } \tl_set:Ne #6 { \fp_eval:n { -(#3) / ( (#1) * (#4) - (#2) * (#3) ) } } \tl_set:Ne #7 { \fp_eval:n { -(#2) / ( (#1) * (#4) - (#2) * (#3) ) } } \tl_set:Ne #8 { \fp_eval:n { (#1) / ( (#1) * (#4) - (#2) * (#3) ) } } } } % Distance from the centre of an axis-aligned rectangle to its boundary along % a ray. The half-extents #2,#3 and result #4 share one coordinate system; % #1 is the ray angle in that system. \cs_new_protected:Npn \__tenkz_geom_rect_ray_reach:nnnN #1#2#3#4 { \fp_compare:nNnTF { cosd(#1) } = {0} { \tl_set:Ne #4 { \fp_eval:n { (#3) / abs(sind(#1)) } } } { \fp_compare:nNnTF { sind(#1) } = {0} { \tl_set:Ne #4 { \fp_eval:n { (#2) / abs(cosd(#1)) } } } { \tl_set:Ne #4 { \fp_eval:n { min( (#2) / abs(cosd(#1)) , (#3) / abs(sind(#1)) ) } } } } } % Update the entry/exit parameters of a ray against one axis-aligned slab. % A zero direction component either leaves that axis unconstrained or proves % that the ray misses the slab. \tl_new:N \l__tenkz_geom_ray_near_tl \tl_new:N \l__tenkz_geom_ray_far_tl \tl_new:N \l__tenkz_geom_ray_enter_tl \tl_new:N \l__tenkz_geom_ray_exit_tl \tl_new:N \l__tenkz_geom_ray_t_tl \bool_new:N \l__tenkz_geom_ray_hit_bool \cs_new_protected:Npn \__tenkz_geom_ray_slab:nnnn #1#2#3#4 { \fp_compare:nNnTF {#2} = {0} { \bool_lazy_or:nnT { \fp_compare_p:nNn {#1} < {#3} } { \fp_compare_p:nNn {#1} > {#4} } { \bool_set_false:N \l__tenkz_geom_ray_hit_bool } } { \tl_set:Ne \l__tenkz_geom_ray_near_tl { \fp_eval:n { min(((#3) - (#1)) / (#2), ((#4) - (#1)) / (#2)) } } \tl_set:Ne \l__tenkz_geom_ray_far_tl { \fp_eval:n { max(((#3) - (#1)) / (#2), ((#4) - (#1)) / (#2)) } } \tl_if_empty:NTF \l__tenkz_geom_ray_enter_tl { \tl_set_eq:NN \l__tenkz_geom_ray_enter_tl \l__tenkz_geom_ray_near_tl } { \tl_set:Ne \l__tenkz_geom_ray_enter_tl { \fp_eval:n { max( \l__tenkz_geom_ray_enter_tl , \l__tenkz_geom_ray_near_tl ) } } } \tl_if_empty:NTF \l__tenkz_geom_ray_exit_tl { \tl_set_eq:NN \l__tenkz_geom_ray_exit_tl \l__tenkz_geom_ray_far_tl } { \tl_set:Ne \l__tenkz_geom_ray_exit_tl { \fp_eval:n { min( \l__tenkz_geom_ray_exit_tl , \l__tenkz_geom_ray_far_tl ) } } } } } % The first forward intersection of a ray with an axis-aligned rectangle. % Positions and bounds are expressed in one local coordinate system; #3 is % the ray angle in that system. An interior origin returns its first exit; % an exterior origin returns its first entry. Both outputs are empty when % the forward ray misses the rectangle. \cs_new_protected:Npn \__tenkz_geom_rect_ray_intersect:nnnnnnnNN #1#2#3#4#5#6#7#8#9 { \bool_set_true:N \l__tenkz_geom_ray_hit_bool \tl_clear:N \l__tenkz_geom_ray_enter_tl \tl_clear:N \l__tenkz_geom_ray_exit_tl \tl_clear:N \l__tenkz_geom_ray_t_tl \__tenkz_geom_ray_slab:nnnn {#1}{cosd(#3)}{#4}{#5} \__tenkz_geom_ray_slab:nnnn {#2}{sind(#3)}{#6}{#7} \bool_if:NT \l__tenkz_geom_ray_hit_bool { \bool_lazy_or:nnT { \fp_compare_p:nNn { \l__tenkz_geom_ray_enter_tl } > { \l__tenkz_geom_ray_exit_tl } } { \fp_compare_p:nNn { \l__tenkz_geom_ray_exit_tl } < {0} } { \bool_set_false:N \l__tenkz_geom_ray_hit_bool } } \bool_if:NTF \l__tenkz_geom_ray_hit_bool { \tl_set:Ne \l__tenkz_geom_ray_t_tl { \fp_eval:n { \l__tenkz_geom_ray_enter_tl >= 0 ? \l__tenkz_geom_ray_enter_tl : \l__tenkz_geom_ray_exit_tl } } \tl_set:Ne #8 { \fp_eval:n { (#1) + \l__tenkz_geom_ray_t_tl * cosd(#3) } } \tl_set:Ne #9 { \fp_eval:n { (#2) + \l__tenkz_geom_ray_t_tl * sind(#3) } } } { \tl_clear:N #8 \tl_clear:N #9 } } % The local east/north basis supplied by one frame address. Affine frames % use E=(b,d), N=(-a,-c); circle frames retain their orthonormal tangent and % outward-normal basis at the named station. \cs_new_protected:Npn \__tenkz_geom_local_basis:nnnNNNN #1#2#3#4#5#6#7 { \prop_get:NnN \l__tenkz_geom_framekind_prop {#1} \l__tenkz_geom_kind_tl \quark_if_no_value:NTF \l__tenkz_geom_kind_tl { \msg_error:nnn {tenkz}{geom-unknown-frame} {#1} \tl_set:Nn #4 {1} \tl_set:Nn #5 {0} \tl_set:Nn #6 {0} \tl_set:Nn #7 {1} } { \str_if_eq:VnTF \l__tenkz_geom_kind_tl {circle} { \__tenkz_geom_turn:nnnN {#1}{#2}{#3} \l__tenkz_geom_tmp_tl \tl_set:Ne #4 { \fp_eval:n { cosd(\l__tenkz_geom_tmp_tl) } } \tl_set:Ne #5 { \fp_eval:n { sind(\l__tenkz_geom_tmp_tl) } } \tl_set:Ne #6 { \fp_eval:n { -sind(\l__tenkz_geom_tmp_tl) } } \tl_set:Ne #7 { \fp_eval:n { cosd(\l__tenkz_geom_tmp_tl) } } } { \tl_set:Ne #4 { \__tenkz_geom_field:nn{#1}{b} } \tl_set:Ne #5 { \__tenkz_geom_field:nn{#1}{d} } \tl_set:Ne #6 { \fp_eval:n { -\__tenkz_geom_field:nn{#1}{a} } } \tl_set:Ne #7 { \fp_eval:n { -\__tenkz_geom_field:nn{#1}{c} } } } } } % Smallest conservative centre-bearing default-bead support margin between % distinct basis-member families in one finite affine cell window. Geometry % owns the exhaustive silhouette calculation, while its caller supplies the % bead-body radius, page-owned gap, and an opaque model-realization callback. % Member offsets are already expressed in the caller's local east/north units. With % dr/dc equal to member j's cell minus % member i's cell, their local displacement is % east = dc + east_j - east_i, % north = -dr + north_j - north_i. % For a nonzero centre vector, the required separation is the support of both % affine-carried beads in that direction plus the page-space gap. At exact % coincidence there is no direction, so the largest affine body support % defines a conservative direction-free convention. A live authored span can % have canonical row or column zero while still intersecting the declared % 1..rows by 1..cols population window. The complete translation domain is % therefore -rows..rows by -cols..cols, one step wider on every side than the % differences between ordinary in-window cells. The strict minimum margin and % fixed traversal order make the reported witness deterministic. % The output is empty when no realized candidate has a negative margin. \prop_new:N \l__tenkz_geom_basis_spacing_east_prop \prop_new:N \l__tenkz_geom_basis_spacing_north_prop \prop_new:N \l__tenkz_geom_basis_spacing_result_prop \int_new:N \l__tenkz_geom_basis_spacing_rows_int \int_new:N \l__tenkz_geom_basis_spacing_cols_int \int_new:N \l__tenkz_geom_basis_spacing_count_int \int_new:N \l__tenkz_geom_basis_spacing_a_int \int_new:N \l__tenkz_geom_basis_spacing_b_int \int_new:N \l__tenkz_geom_basis_spacing_dr_int \int_new:N \l__tenkz_geom_basis_spacing_dc_int \tl_new:N \l__tenkz_geom_basis_spacing_radius_tl \tl_new:N \l__tenkz_geom_basis_spacing_gap_tl \tl_new:N \l__tenkz_geom_basis_spacing_ex_tl \tl_new:N \l__tenkz_geom_basis_spacing_ey_tl \tl_new:N \l__tenkz_geom_basis_spacing_nx_tl \tl_new:N \l__tenkz_geom_basis_spacing_ny_tl \tl_new:N \l__tenkz_geom_basis_spacing_east_a_tl \tl_new:N \l__tenkz_geom_basis_spacing_east_b_tl \tl_new:N \l__tenkz_geom_basis_spacing_north_a_tl \tl_new:N \l__tenkz_geom_basis_spacing_north_b_tl \tl_new:N \l__tenkz_geom_basis_spacing_east_delta_tl \tl_new:N \l__tenkz_geom_basis_spacing_north_delta_tl \tl_new:N \l__tenkz_geom_basis_spacing_x_tl \tl_new:N \l__tenkz_geom_basis_spacing_y_tl \tl_new:N \l__tenkz_geom_basis_spacing_dist_square_tl \tl_new:N \l__tenkz_geom_basis_spacing_dist_tl \tl_new:N \l__tenkz_geom_basis_spacing_floor_tl \tl_new:N \l__tenkz_geom_basis_spacing_clearance_tl \bool_new:N \l__tenkz_geom_basis_spacing_kind_ok_bool \cs_new_protected:Npn \__tenkz_geom_basis_spacing_query:nnnnnTF #1#2#3#4#5#6#7 {#7} % Largest stretch of a complete two-dimensional basis. This is the largest % singular value, written from the trace and determinant of M^T M so exact % centre coincidence has a direction-independent bead support. \cs_new:Npn \__tenkz_geom_basis_max_stretch:nnnn #1#2#3#4 { sqrt( ( (#1)^2 + (#2)^2 + (#3)^2 + (#4)^2 + sqrt( max( 0, ( (#1)^2 + (#2)^2 + (#3)^2 + (#4)^2 )^2 - 4 * ( (#1) * (#4) - (#2) * (#3) )^2 ) ) ) / 2 ) } \cs_new_protected:Npn \__tenkz_geom_basis_spacing_save: { \prop_put:Nne \l__tenkz_geom_basis_spacing_result_prop {member-a} { \int_use:N \l__tenkz_geom_basis_spacing_a_int } \prop_put:Nne \l__tenkz_geom_basis_spacing_result_prop {member-b} { \int_use:N \l__tenkz_geom_basis_spacing_b_int } \prop_put:Nne \l__tenkz_geom_basis_spacing_result_prop {dr} { \int_use:N \l__tenkz_geom_basis_spacing_dr_int } \prop_put:Nne \l__tenkz_geom_basis_spacing_result_prop {dc} { \int_use:N \l__tenkz_geom_basis_spacing_dc_int } \prop_put:Nne \l__tenkz_geom_basis_spacing_result_prop {dist} { \l__tenkz_geom_basis_spacing_dist_tl } \prop_put:Nne \l__tenkz_geom_basis_spacing_result_prop {floor} { \l__tenkz_geom_basis_spacing_floor_tl } \prop_put:Nne \l__tenkz_geom_basis_spacing_result_prop {clearance} { \l__tenkz_geom_basis_spacing_clearance_tl } } \cs_new_protected:Npn \__tenkz_geom_basis_spacing_save_if_realized: { \__tenkz_geom_basis_spacing_query:nnnnnTF {realized} { \int_use:N \l__tenkz_geom_basis_spacing_a_int } { \int_use:N \l__tenkz_geom_basis_spacing_b_int } { \int_use:N \l__tenkz_geom_basis_spacing_dr_int } { \int_use:N \l__tenkz_geom_basis_spacing_dc_int } { \__tenkz_geom_basis_spacing_save: } { } } \cs_new_protected:Npn \__tenkz_geom_basis_spacing_candidate_measure: { \__tenkz_geom_basis_apply:nnnnnnNN { \l__tenkz_geom_basis_spacing_ex_tl } { \l__tenkz_geom_basis_spacing_ey_tl } { \l__tenkz_geom_basis_spacing_nx_tl } { \l__tenkz_geom_basis_spacing_ny_tl } { \l__tenkz_geom_basis_spacing_dc_int + \l__tenkz_geom_basis_spacing_east_delta_tl } { -\l__tenkz_geom_basis_spacing_dr_int + \l__tenkz_geom_basis_spacing_north_delta_tl } \l__tenkz_geom_basis_spacing_x_tl \l__tenkz_geom_basis_spacing_y_tl \tl_set:Ne \l__tenkz_geom_basis_spacing_dist_square_tl { \fp_eval:n { ( \l__tenkz_geom_basis_spacing_x_tl ) ^ 2 + ( \l__tenkz_geom_basis_spacing_y_tl ) ^ 2 } } \fp_compare:nNnTF { \l__tenkz_geom_basis_spacing_dist_square_tl } = {0} { \tl_set:Nn \l__tenkz_geom_basis_spacing_dist_tl {0} \tl_set:Ne \l__tenkz_geom_basis_spacing_floor_tl { \fp_eval:n { \l__tenkz_geom_basis_spacing_gap_tl + 2 * \l__tenkz_geom_basis_spacing_radius_tl * \__tenkz_geom_basis_max_stretch:nnnn { \l__tenkz_geom_basis_spacing_ex_tl } { \l__tenkz_geom_basis_spacing_ey_tl } { \l__tenkz_geom_basis_spacing_nx_tl } { \l__tenkz_geom_basis_spacing_ny_tl } } } } { \tl_set:Ne \l__tenkz_geom_basis_spacing_dist_tl { \fp_eval:n { sqrt(\l__tenkz_geom_basis_spacing_dist_square_tl) } } \tl_set:Ne \l__tenkz_geom_basis_spacing_floor_tl { \fp_eval:n { \l__tenkz_geom_basis_spacing_gap_tl + 2 * \__tenkz_geom_support_basis:nnnnnnnnn {circle} { \l__tenkz_geom_basis_spacing_radius_tl } { \l__tenkz_geom_basis_spacing_radius_tl } {0} { \l__tenkz_geom_basis_spacing_ex_tl } { \l__tenkz_geom_basis_spacing_ey_tl } { \l__tenkz_geom_basis_spacing_nx_tl } { \l__tenkz_geom_basis_spacing_ny_tl } { atand( \l__tenkz_geom_basis_spacing_y_tl, \l__tenkz_geom_basis_spacing_x_tl ) } } } } \tl_set:Ne \l__tenkz_geom_basis_spacing_clearance_tl { \fp_eval:n { \l__tenkz_geom_basis_spacing_dist_tl - \l__tenkz_geom_basis_spacing_floor_tl } } \fp_compare:nNnT { \l__tenkz_geom_basis_spacing_clearance_tl } < {0} { \prop_if_empty:NTF \l__tenkz_geom_basis_spacing_result_prop { \__tenkz_geom_basis_spacing_save_if_realized: } { \fp_compare:nNnT { \l__tenkz_geom_basis_spacing_clearance_tl } < { \prop_item:Nn \l__tenkz_geom_basis_spacing_result_prop {clearance} } { \__tenkz_geom_basis_spacing_save_if_realized: } } } } \cs_new_protected:Npn \__tenkz_geom_basis_spacing_candidate: { \__tenkz_geom_basis_spacing_query:nnnnnTF {candidate} { \int_use:N \l__tenkz_geom_basis_spacing_a_int } { \int_use:N \l__tenkz_geom_basis_spacing_b_int } { \int_use:N \l__tenkz_geom_basis_spacing_dr_int } { \int_use:N \l__tenkz_geom_basis_spacing_dc_int } { \__tenkz_geom_basis_spacing_candidate_measure: } { } } \cs_new_protected:Npn \__tenkz_geom_basis_spacing_dc:n #1 { \int_set:Nn \l__tenkz_geom_basis_spacing_dc_int {#1} \__tenkz_geom_basis_spacing_candidate: } \cs_new_protected:Npn \__tenkz_geom_basis_spacing_dr:n #1 { \int_set:Nn \l__tenkz_geom_basis_spacing_dr_int {#1} \int_step_function:nnN { -\l__tenkz_geom_basis_spacing_cols_int } { \l__tenkz_geom_basis_spacing_cols_int } \__tenkz_geom_basis_spacing_dc:n } \cs_new_protected:Npn \__tenkz_geom_basis_spacing_b:n #1 { \int_set:Nn \l__tenkz_geom_basis_spacing_b_int {#1} \prop_get:NeN \l__tenkz_geom_basis_spacing_east_prop {#1} \l__tenkz_geom_basis_spacing_east_b_tl \prop_get:NeN \l__tenkz_geom_basis_spacing_north_prop {#1} \l__tenkz_geom_basis_spacing_north_b_tl \tl_set:Ne \l__tenkz_geom_basis_spacing_east_delta_tl { \fp_eval:n { \l__tenkz_geom_basis_spacing_east_b_tl - \l__tenkz_geom_basis_spacing_east_a_tl } } \tl_set:Ne \l__tenkz_geom_basis_spacing_north_delta_tl { \fp_eval:n { \l__tenkz_geom_basis_spacing_north_b_tl - \l__tenkz_geom_basis_spacing_north_a_tl } } \int_step_function:nnN { -\l__tenkz_geom_basis_spacing_rows_int } { \l__tenkz_geom_basis_spacing_rows_int } \__tenkz_geom_basis_spacing_dr:n } \cs_new_protected:Npn \__tenkz_geom_basis_spacing_a:n #1 { \int_set:Nn \l__tenkz_geom_basis_spacing_a_int {#1} \prop_get:NeN \l__tenkz_geom_basis_spacing_east_prop {#1} \l__tenkz_geom_basis_spacing_east_a_tl \prop_get:NeN \l__tenkz_geom_basis_spacing_north_prop {#1} \l__tenkz_geom_basis_spacing_north_a_tl \int_step_function:nnN { \l__tenkz_geom_basis_spacing_a_int + 1 } { \l__tenkz_geom_basis_spacing_count_int } \__tenkz_geom_basis_spacing_b:n } \cs_new_protected:Npn \__tenkz_geom_basis_spacing:nnnnnNNNN #1#2#3#4#5#6#7#8#9 { \prop_clear:N #9 \prop_clear:N \l__tenkz_geom_basis_spacing_result_prop \__tenkz_geom_require_kind:nnnN {#1} {affine} {basis-spacing} \l__tenkz_geom_basis_spacing_kind_ok_bool \bool_if:NT \l__tenkz_geom_basis_spacing_kind_ok_bool { \int_set:Nn \l__tenkz_geom_basis_spacing_rows_int {#2} \int_set:Nn \l__tenkz_geom_basis_spacing_cols_int {#3} \tl_set:Ne \l__tenkz_geom_basis_spacing_radius_tl { \fp_eval:n {#4} } \tl_set:Ne \l__tenkz_geom_basis_spacing_gap_tl { \fp_eval:n {#5} } \prop_set_eq:NN \l__tenkz_geom_basis_spacing_east_prop #6 \prop_set_eq:NN \l__tenkz_geom_basis_spacing_north_prop #7 \cs_set_eq:NN \__tenkz_geom_basis_spacing_query:nnnnnTF #8 \int_set:Nn \l__tenkz_geom_basis_spacing_count_int { \prop_count:N \l__tenkz_geom_basis_spacing_east_prop } \__tenkz_geom_local_basis:nnnNNNN {#1}{1}{1} \l__tenkz_geom_basis_spacing_ex_tl \l__tenkz_geom_basis_spacing_ey_tl \l__tenkz_geom_basis_spacing_nx_tl \l__tenkz_geom_basis_spacing_ny_tl \int_step_function:nnN {1} { \l__tenkz_geom_basis_spacing_count_int - 1 } \__tenkz_geom_basis_spacing_a:n \prop_set_eq:NN #9 \l__tenkz_geom_basis_spacing_result_prop } } % The page bearing of one direction in the local east/north axes of a frame % address. Both affine and circular carriers answer through the same complete % basis; consumers do not inspect frame kinds or reproduce projection formulae. \tl_new:N \l__tenkz_geom_bearing_ex_tl \tl_new:N \l__tenkz_geom_bearing_ey_tl \tl_new:N \l__tenkz_geom_bearing_nx_tl \tl_new:N \l__tenkz_geom_bearing_ny_tl \cs_new_protected:Npn \__tenkz_geom_local_bearing:nnnnN #1#2#3#4#5 { \__tenkz_geom_local_basis:nnnNNNN {#1}{#2}{#3} \l__tenkz_geom_bearing_ex_tl \l__tenkz_geom_bearing_ey_tl \l__tenkz_geom_bearing_nx_tl \l__tenkz_geom_bearing_ny_tl \__tenkz_geom_basis_bearing:nnnnnN { \l__tenkz_geom_bearing_ex_tl } { \l__tenkz_geom_bearing_ey_tl } { \l__tenkz_geom_bearing_nx_tl } { \l__tenkz_geom_bearing_ny_tl } {#4} #5 } % The axes a frame gives a record. An address on a carrier is read in the % carrier's axes, and the ink the frame measures is measured in them, so % every stage that turns ink or resolves a face asks this one question: by % what angle does the frame turn the axes at this address? For an affine % frame the answer is its linear part -- the image of the logical direction % the flat frame sends to page east, the same everywhere -- and for a circle % frame it is the tangent, which varies by station. A shear is not a turn: % under an affine frame that scales its axes unequally this angle is the % turn of one axis only, and the other follows the matrix. \cs_new_protected:Npn \__tenkz_geom_turn:nnnN #1#2#3#4 { \prop_get:NnN \l__tenkz_geom_framekind_prop {#1} \l__tenkz_geom_kind_tl \quark_if_no_value:NTF \l__tenkz_geom_kind_tl { \msg_error:nnn {tenkz}{geom-unknown-frame} {#1} \tl_set:Nn #4 {0} } { \str_if_eq:VnTF \l__tenkz_geom_kind_tl {circle} { \tl_set:Ne #4 { \fp_eval:n { \__tenkz_geom_station_angle:nn {#1} {#3} - 90 } } } { \tl_set:Ne #4 { \fp_eval:n { atand( \__tenkz_geom_field:nn{#1}{d} , \__tenkz_geom_field:nn{#1}{b} ) } } } } } % compass faces convert once, at the parse boundary \cs_new:Npn \__tenkz_geom_compass_angle:n #1 { \str_case:nnF {#1} { {n}{90} {e}{0} {s}{270} {w}{180} } { \msg_expandable_error:nnn {tenkz}{geom-compass} {#1} 0 } } \cs_new_protected:Npn \__tenkz_geom_port_angle:nnN #1#2#3 { \__tenkz_geom_dir:nnN {#1} { \__tenkz_geom_compass_angle:n {#2} } #3 } % station k of a circle frame: points sit clockwise from `start`, and the % travel tangent trails the radius by a quarter turn \cs_new:Npn \__tenkz_geom_station_angle:nn #1#2 { \__tenkz_geom_field:nn{#1}{start} - 360 * ( (#2) - 1 ) / \__tenkz_geom_field:nn{#1}{n} } \cs_new_protected:Npn \__tenkz_geom_station:nnNNN #1#2#3#4#5 { \tl_clear:N #3 \tl_clear:N #4 \tl_clear:N #5 \__tenkz_geom_require_kind:nnnN {#1} {circle} {station} \l__tenkz_geom_kind_ok_bool \bool_if:NT \l__tenkz_geom_kind_ok_bool { \tl_set:Ne \l__tenkz_geom_tmp_tl { \fp_eval:n { \__tenkz_geom_station_angle:nn {#1} {#2} } } \tl_set:Ne #3 { \fp_eval:n { \__tenkz_geom_field:nn{#1}{r} * cosd(\l__tenkz_geom_tmp_tl) } } \tl_set:Ne #4 { \fp_eval:n { \__tenkz_geom_field:nn{#1}{r} * sind(\l__tenkz_geom_tmp_tl) } } \tl_set:Ne #5 { \fp_eval:n { \l__tenkz_geom_tmp_tl - 90 } } } } % ---------- the placement graph --------------------------------------------- % A request is stored unresolved and evaluated on demand; evaluation walks % dependencies depth-first, marks the path, and a revisit names the cycle. \__tenkz_prop_new_indexed:N \l__tenkz_geom_req_prop % id -> {type}{arg}{arg}{arg} \__tenkz_prop_new_indexed:N \l__tenkz_geom_x_prop % id -> fp literal (pitch units) \__tenkz_prop_new_indexed:N \l__tenkz_geom_y_prop \__tenkz_prop_new_indexed:N \l__tenkz_geom_state_prop % id -> visiting \seq_new:N \l__tenkz_geom_trail_seq \tl_new:N \l__tenkz_geom_xa_tl \tl_new:N \l__tenkz_geom_ya_tl \bool_new:N \l__tenkz_geom_ok_bool \cs_new_protected:Npn \__tenkz_geom_reset: { \prop_clear:N \l__tenkz_geom_frame_prop \prop_clear:N \l__tenkz_geom_framekind_prop \prop_clear:N \l__tenkz_geom_req_prop \prop_clear:N \l__tenkz_geom_x_prop \prop_clear:N \l__tenkz_geom_y_prop \prop_clear:N \l__tenkz_geom_state_prop \seq_clear:N \l__tenkz_geom_trail_seq } \cs_new_protected:Npn \__tenkz_geom_place_cell:nnnn #1#2#3#4 { \prop_put:Nnn \l__tenkz_geom_req_prop {#1} { {cell}{#2}{#3}{#4} } } \cs_generate_variant:Nn \__tenkz_geom_place_cell:nnnn { nnee } \cs_new_protected:Npn \__tenkz_geom_place_rel:nnnn #1#2#3#4 { \prop_put:Nnn \l__tenkz_geom_req_prop {#1} { {rel}{#2}{#3}{#4} } } \cs_generate_variant:Nn \__tenkz_geom_place_rel:nnnn { nnee, nVee, neee } \cs_new_protected:Npn \__tenkz_geom_place_offset:nnnn #1#2#3#4 { \prop_put:Nnn \l__tenkz_geom_req_prop {#1} { {offset}{#2}{#3}{#4} } } \cs_generate_variant:Nn \__tenkz_geom_place_offset:nnnn { nVee } \cs_new_protected:Npn \__tenkz_geom_place_mid:nnn #1#2#3 { \prop_put:Nnn \l__tenkz_geom_req_prop {#1} { {mid}{#2}{#3}{} } } \cs_generate_variant:Nn \__tenkz_geom_place_mid:nnn { nVV, nee } \cs_new_protected:Npn \__tenkz_geom_place_onseg:nnnn #1#2#3#4 { \prop_put:Nnn \l__tenkz_geom_req_prop {#1} { {onseg}{#2}{#3}{#4} } } \cs_new_protected:Npn \__tenkz_geom_store:nnn #1#2#3 { \prop_put:Nne \l__tenkz_geom_x_prop {#1} { \fp_eval:n {#2} } \prop_put:Nne \l__tenkz_geom_y_prop {#1} { \fp_eval:n {#3} } \prop_remove:Nn \l__tenkz_geom_state_prop {#1} } \cs_new_protected:Npn \__tenkz_geom_need:nN #1#2 { \bool_set_false:N #2 \prop_if_in:NnTF \l__tenkz_geom_x_prop {#1} { \bool_set_true:N #2 } { \prop_if_in:NnTF \l__tenkz_geom_state_prop {#1} { \msg_error:nne {tenkz}{geom-place-cycle} { \seq_use:Nn \l__tenkz_geom_trail_seq { ~->~ } ~->~ #1 } } { \prop_put:Nnn \l__tenkz_geom_state_prop {#1} {visiting} \seq_put_right:Nn \l__tenkz_geom_trail_seq {#1} \prop_get:NnN \l__tenkz_geom_req_prop {#1} \l__tenkz_geom_tmp_tl \quark_if_no_value:NTF \l__tenkz_geom_tmp_tl { \msg_error:nnn {tenkz}{geom-unknown-place} {#1} } { \exp_last_unbraced:NV \__tenkz_geom_eval:nnnnnN \l__tenkz_geom_tmp_tl {#1} #2 } \prop_remove:Nn \l__tenkz_geom_state_prop {#1} \seq_pop_right:NN \l__tenkz_geom_trail_seq \l__tenkz_geom_tmp_tl } } } \cs_new_protected:Npn \__tenkz_geom_need:n #1 { \__tenkz_geom_need:nN {#1} \l__tenkz_geom_ok_bool } % #1 type, #2-#4 request args, #5 the id being resolved, #6 success \cs_new_protected:Npn \__tenkz_geom_eval:nnnnnN #1#2#3#4#5#6 { \str_case:nn {#1} { {cell} { \__tenkz_geom_apply:nnnNNN {#2} {#3} {#4} \l__tenkz_geom_xa_tl \l__tenkz_geom_ya_tl #6 \bool_if:NT #6 { \__tenkz_geom_store:nnn {#5} { \l__tenkz_geom_xa_tl } { \l__tenkz_geom_ya_tl } } } {rel} { \__tenkz_geom_need:nN {#2} #6 \bool_if:NT #6 { \__tenkz_geom_store:nnn {#5} { \prop_item:Nn \l__tenkz_geom_x_prop {#2} + (#4) * cosd(#3) } { \prop_item:Nn \l__tenkz_geom_y_prop {#2} + (#4) * sind(#3) } } } {offset} { \__tenkz_geom_need:nN {#2} #6 \bool_if:NT #6 { \__tenkz_geom_store:nnn {#5} { \prop_item:Nn \l__tenkz_geom_x_prop {#2} + (#3) } { \prop_item:Nn \l__tenkz_geom_y_prop {#2} + (#4) } } } {mid} { \__tenkz_geom_need:nN {#2} #6 \bool_if:NT #6 { \__tenkz_geom_need:nN {#3} #6 } \bool_if:NT #6 { \__tenkz_geom_store:nnn {#5} { ( \prop_item:Nn \l__tenkz_geom_x_prop {#2} + \prop_item:Nn \l__tenkz_geom_x_prop {#3} ) / 2 } { ( \prop_item:Nn \l__tenkz_geom_y_prop {#2} + \prop_item:Nn \l__tenkz_geom_y_prop {#3} ) / 2 } } } {onseg} { \__tenkz_geom_need:nN {#2} #6 \bool_if:NT #6 { \__tenkz_geom_need:nN {#3} #6 } \bool_if:NT #6 { \__tenkz_geom_store:nnn {#5} { \prop_item:Nn \l__tenkz_geom_x_prop {#2} + (#4) * ( \prop_item:Nn \l__tenkz_geom_x_prop {#3} - \prop_item:Nn \l__tenkz_geom_x_prop {#2} ) } { \prop_item:Nn \l__tenkz_geom_y_prop {#2} + (#4) * ( \prop_item:Nn \l__tenkz_geom_y_prop {#3} - \prop_item:Nn \l__tenkz_geom_y_prop {#2} ) } } } } } \cs_new_protected:Npn \__tenkz_geom_resolve: { \prop_map_inline:Nn \l__tenkz_geom_req_prop { \__tenkz_geom_need:n {##1} } } \cs_new_protected:Npn \__tenkz_geom_xy:nNN #1#2#3 { \__tenkz_geom_need:nN {#1} \l__tenkz_geom_ok_bool \bool_if:NTF \l__tenkz_geom_ok_bool { \prop_get:NnN \l__tenkz_geom_x_prop {#1} #2 \prop_get:NnN \l__tenkz_geom_y_prop {#1} #3 } { \tl_clear:N #2 \tl_clear:N #3 } } % ---------- silhouettes as support functions -------------------------------- % Wire-endpoint contract for the coming weight vocabulary: each strand of a % double/bundle wire trims against the support ALONG ITS OWN OFFSET LINE, % never the centerline -- for a circle of radius r and lateral offset g the % endpoint sits at (+-g, -sqrt(r^2 - g^2)). The support query answers this % generally; no per-shape anchor tables. % Support of a skin rotated by theta, queried along direction d, in pitch % units. Arguments are family, half-extents, corner radius, theta, and d. % A roundrect caller supplies the radius of the live rendered skin, normalized % to pitch units; the other families ignore that slot. Families: circle % (hx = radius), rect (half-extents), roundrect (rect shrunk by its actual % corner radius, then re-inflated), triangle (isoceles, apex east: vertex max % is exact for polygons), triwest (the same isoceles triangle turned half a % circle about its own centre, so its apex points west). The half turn is % one sign: the apex stands at +hx on the record's own east axis and the two % base corners at -hx, and the triangle is symmetric about that axis, so % reading the established formula with the apex extent negated IS the % mirrored silhouette. One formula therefore serves both orientations, in % the turned record here and in the affine record below. \cs_new:Npn \__tenkz_geom_support:nnnnnn #1#2#3#4#5#6 { \str_case:nnF {#1} { {circle} { (#2) } {rect} { abs( (#2) * cosd((#6)-(#5)) ) + abs( (#3) * sind((#6)-(#5)) ) } {roundrect} { abs( ( (#2) - \__tenkz_geom_rr:nnn{#4}{#2}{#3} ) * cosd((#6)-(#5)) ) + abs( ( (#3) - \__tenkz_geom_rr:nnn{#4}{#2}{#3} ) * sind((#6)-(#5)) ) + \__tenkz_geom_rr:nnn{#4}{#2}{#3} } {triangle} { max( (#2) * cosd((#6)-(#5)) , - (#2) * cosd((#6)-(#5)) + (#3) * sind((#6)-(#5)) , - (#2) * cosd((#6)-(#5)) - (#3) * sind((#6)-(#5)) ) } {triwest} { \__tenkz_geom_support:nnnnnn {triangle}{-(#2)}{#3}{#4}{#5}{#6} } } { \msg_expandable_error:nnn {tenkz}{geom-family} {#1} 0 } } \cs_new:Npn \__tenkz_geom_rr:nnn #1#2#3 { min( (#1) , (#2) , (#3) ) } % Radial reach of a rounded rectangle from its centre along bearing #4: the % distance at which the outgoing ray meets the boundary. This is the % port-trim query, distinct from the support fold above -- a support in a % direction is attained off-ray at a corner, while a wire endpoint stands ON % the ray. A flat-side hit divides the half-extent by its direction cosine; % a cap hit solves the ray's quadratic against the corner disc. Half-extents % #1, #2 and corner radius #3 share one unit; the reach lands in #5 in that % unit. The side tests keep every division away from a vanishing cosine: a % ray parallel to a side can only leave through the other pair of sides or a % cap. \fp_new:N \l__tenkz_geom_ray_u_fp \fp_new:N \l__tenkz_geom_ray_v_fp \fp_new:N \l__tenkz_geom_ray_r_fp \fp_new:N \l__tenkz_geom_ray_cx_fp \fp_new:N \l__tenkz_geom_ray_cy_fp \fp_new:N \l__tenkz_geom_ray_b_fp \cs_new_protected:Npn \__tenkz_geom_roundrect_ray:nnnnN #1#2#3#4#5 { \fp_set:Nn \l__tenkz_geom_ray_u_fp { abs( cosd(#4) ) } \fp_set:Nn \l__tenkz_geom_ray_v_fp { abs( sind(#4) ) } \fp_set:Nn \l__tenkz_geom_ray_r_fp { \__tenkz_geom_rr:nnn {#3}{#1}{#2} } \fp_set:Nn \l__tenkz_geom_ray_cx_fp { (#1) - \l__tenkz_geom_ray_r_fp } \fp_set:Nn \l__tenkz_geom_ray_cy_fp { (#2) - \l__tenkz_geom_ray_r_fp } \fp_compare:nNnTF { (#1) * \l__tenkz_geom_ray_v_fp } > { \l__tenkz_geom_ray_cy_fp * \l__tenkz_geom_ray_u_fp } { \fp_compare:nNnTF { (#2) * \l__tenkz_geom_ray_u_fp } > { \l__tenkz_geom_ray_cx_fp * \l__tenkz_geom_ray_v_fp } { \fp_set:Nn \l__tenkz_geom_ray_b_fp { \l__tenkz_geom_ray_cx_fp * \l__tenkz_geom_ray_u_fp + \l__tenkz_geom_ray_cy_fp * \l__tenkz_geom_ray_v_fp } \fp_set:Nn #5 { \l__tenkz_geom_ray_b_fp + sqrt( max( 0 , \l__tenkz_geom_ray_b_fp ^ 2 - \l__tenkz_geom_ray_cx_fp ^ 2 - \l__tenkz_geom_ray_cy_fp ^ 2 + \l__tenkz_geom_ray_r_fp ^ 2 ) ) } } { \fp_set:Nn #5 { (#2) / \l__tenkz_geom_ray_v_fp } } } { \fp_set:Nn #5 { (#1) / \l__tenkz_geom_ray_u_fp } } } % Support of one primitive transported through a complete east/north basis. % This is a distinct contract from the scalar-turn function above: an affine % support record is % {affine}{family}{hx}{hy}{radius}{ex}{ey}{nx}{ny}{cx}{cy}. % The explicit tag keeps a basis out of the rotated record's angle slot and % lets one obstacle sequence contain established rotated and affine shapes. \cs_new:Npn \__tenkz_geom_support_basis:nnnnnnnnn #1#2#3#4#5#6#7#8#9 { \str_case:enF {#1} { {circle} { (#2) * sqrt( ( (#5) * cosd(#9) + (#6) * sind(#9) ) ^ 2 + ( (#7) * cosd(#9) + (#8) * sind(#9) ) ^ 2 ) } {rect} { abs( (#2) * ( (#5) * cosd(#9) + (#6) * sind(#9) ) ) + abs( (#3) * ( (#7) * cosd(#9) + (#8) * sind(#9) ) ) } {roundrect} { abs( ( (#2) - \__tenkz_geom_rr:nnn{#4}{#2}{#3} ) * ( (#5) * cosd(#9) + (#6) * sind(#9) ) ) + abs( ( (#3) - \__tenkz_geom_rr:nnn{#4}{#2}{#3} ) * ( (#7) * cosd(#9) + (#8) * sind(#9) ) ) + \__tenkz_geom_rr:nnn{#4}{#2}{#3} * sqrt( ( (#5) * cosd(#9) + (#6) * sind(#9) ) ^ 2 + ( (#7) * cosd(#9) + (#8) * sind(#9) ) ^ 2 ) } {triangle} { max( (#2) * ( (#5) * cosd(#9) + (#6) * sind(#9) ) , - (#2) * ( (#5) * cosd(#9) + (#6) * sind(#9) ) + (#3) * ( (#7) * cosd(#9) + (#8) * sind(#9) ) , - (#2) * ( (#5) * cosd(#9) + (#6) * sind(#9) ) - (#3) * ( (#7) * cosd(#9) + (#8) * sind(#9) ) ) } {triwest} { \__tenkz_geom_support_basis:nnnnnnnnn {triangle}{-(#2)}{#3}{#4}{#5}{#6}{#7}{#8}{#9} } } { \msg_expandable_error:nnn {tenkz}{geom-family} {#1} 0 } } % fold max-support over {family}{hx}{hy}{radius}{theta}{cx}{cy} records: % each contributes its centre's reach plus its shape's support \cs_new_protected:Npn \__tenkz_geom_support_max:nnN #1#2#3 { \tl_set:Nn #3 { -10000 } \seq_map_inline:cn {#1} { \__tenkz_geom_support_max_record:nnN {##1} {#2} #3 } } \cs_new_protected:Npn \__tenkz_geom_support_max_record:nnN #1#2#3 { \__tenkz_geom_support_max_record_aux:w #1 \q_stop {#2} #3 } \cs_new_protected:Npn \__tenkz_geom_support_max_record_aux:w #1#2\q_stop #3#4 { \str_if_eq:nnTF {#1}{affine} { \__tenkz_geom_support_max_affine_setup:w #2 \q_stop {#3} #4 } { \__tenkz_geom_support_max_one:nnnnnnnnN {#1} #2 {#3} #4 } } \cs_new_protected:Npn \__tenkz_geom_support_max_one:nnnnnnnnN #1#2#3#4#5#6#7#8#9 { \tl_set:Ne #9 { \fp_eval:n { max( #9 , (#6) * cosd(#8) + (#7) * sind(#8) + \__tenkz_geom_support:nnnnnn {#1}{#2}{#3}{#4}{#5}{#8} ) } } } \tl_new:N \l__tenkz_geom_support_family_tl \tl_new:N \l__tenkz_geom_support_hx_tl \tl_new:N \l__tenkz_geom_support_hy_tl \tl_new:N \l__tenkz_geom_support_radius_tl \tl_new:N \l__tenkz_geom_support_shape_tl \cs_new_protected:Npn \__tenkz_geom_support_max_affine_setup:w #1#2#3#4#5\q_stop #6#7 { \tl_set:Nn \l__tenkz_geom_support_family_tl {#1} \tl_set:Nn \l__tenkz_geom_support_hx_tl {#2} \tl_set:Nn \l__tenkz_geom_support_hy_tl {#3} \tl_set:Nn \l__tenkz_geom_support_radius_tl {#4} \__tenkz_geom_support_max_affine_finish:w #5 \q_stop {#6} #7 } \cs_new_protected:Npn \__tenkz_geom_support_max_affine_finish:w #1#2#3#4#5#6\q_stop #7#8 { \tl_set:Ne #8 { \fp_eval:n { max( #8 , (#5) * cosd(#7) + (#6) * sind(#7) + \__tenkz_geom_support_basis:nnnnnnnnn { \l__tenkz_geom_support_family_tl } { \l__tenkz_geom_support_hx_tl } { \l__tenkz_geom_support_hy_tl } { \l__tenkz_geom_support_radius_tl } {#1}{#2}{#3}{#4}{#7} ) } } } % Fold support against an arbitrary page covector (gx,gy). The result is in % the covector's coordinate units, so a dual-basis row returns an exact local % coordinate even when the carrier is sheared or scaled. \tl_new:N \l__tenkz_geom_support_gx_tl \tl_new:N \l__tenkz_geom_support_gy_tl \cs_new_protected:Npn \__tenkz_geom_support_max_linear:nnnN #1#2#3#4 { \tl_set:Nn \l__tenkz_geom_support_gx_tl {#2} \tl_set:Nn \l__tenkz_geom_support_gy_tl {#3} \tl_set:Nn #4 { -10000 } \seq_map_inline:cn {#1} { \__tenkz_geom_support_max_linear_record:nN {##1} #4 } } \cs_new_protected:Npn \__tenkz_geom_support_max_linear_record:nN #1#2 { \__tenkz_geom_support_max_linear_record_aux:w #1 \q_stop #2 } \cs_new_protected:Npn \__tenkz_geom_support_max_linear_record_aux:w #1#2\q_stop #3 { \str_if_eq:nnTF {#1}{affine} { \__tenkz_geom_support_max_linear_affine_setup:w #2 \q_stop #3 } { \__tenkz_geom_support_max_linear_one:nnnnnnnN {#1} #2 #3 } } \cs_new_protected:Npn \__tenkz_geom_support_max_linear_one:nnnnnnnN #1#2#3#4#5#6#7#8 { \tl_set:Ne #8 { \fp_eval:n { max( #8 , (#6) * \l__tenkz_geom_support_gx_tl + (#7) * \l__tenkz_geom_support_gy_tl + sqrt( ( \l__tenkz_geom_support_gx_tl ) ^ 2 + ( \l__tenkz_geom_support_gy_tl ) ^ 2 ) * \__tenkz_geom_support:nnnnnn {#1}{#2}{#3}{#4}{#5} { atand( \l__tenkz_geom_support_gy_tl , \l__tenkz_geom_support_gx_tl ) } ) } } } \cs_new_protected:Npn \__tenkz_geom_support_max_linear_affine_setup:w #1#2#3#4#5\q_stop #6 { \tl_set:Nn \l__tenkz_geom_support_family_tl {#1} \tl_set:Nn \l__tenkz_geom_support_hx_tl {#2} \tl_set:Nn \l__tenkz_geom_support_hy_tl {#3} \tl_set:Nn \l__tenkz_geom_support_radius_tl {#4} \__tenkz_geom_support_max_linear_affine_finish:w #5 \q_stop #6 } \cs_new_protected:Npn \__tenkz_geom_support_max_linear_affine_finish:w #1#2#3#4#5#6\q_stop #7 { \tl_set:Ne \l__tenkz_geom_support_shape_tl { \fp_eval:n { \__tenkz_geom_support_basis:nnnnnnnnn { \l__tenkz_geom_support_family_tl } { \l__tenkz_geom_support_hx_tl } { \l__tenkz_geom_support_hy_tl } { \l__tenkz_geom_support_radius_tl } {#1}{#2}{#3}{#4} { atand( \l__tenkz_geom_support_gy_tl , \l__tenkz_geom_support_gx_tl ) } } } \tl_set:Ne #7 { \fp_eval:n { max( #7 , (#5) * \l__tenkz_geom_support_gx_tl + (#6) * \l__tenkz_geom_support_gy_tl + sqrt( ( \l__tenkz_geom_support_gx_tl ) ^ 2 + ( \l__tenkz_geom_support_gy_tl ) ^ 2 ) * \l__tenkz_geom_support_shape_tl ) } } } % ---------- probes ----------------------------------------------------------- % Test fixtures assert geometry through the event stream; six decimals keep % ordinary coordinates stable. A signed decision margin retains nine so a % strict comparison remains visible when its two displayed operands coincide. \cs_new_protected:Npn \__tenkz_geom_probe:nn #1#2 { \tenkz@event{geomprobe|id=#1|#2} } \cs_new:Npn \__tenkz_geom_round:n #1 { \fp_eval:n { round( #1 , 6 ) } } \cs_new:Npn \__tenkz_geom_round_margin:n #1 { \fp_eval:n { round( #1 , 9 ) } } \ExplSyntaxOff \endinput