More work on DCEL correctness

lightningbeam-ui/lightningbeam-core/src/clipboard.rs

@@ -123,13 +123,13 @@ pub enum ClipboardContent {
 
     /// Selected DCEL geometry from a vector layer.
     ///
-    /// Currently a stub — `data` is opaque bytes whose format is TBD in Phase 2
-    /// once DCEL serialization is implemented.  Copy/paste of vector shapes does
-    /// nothing until then.  Secondary formats (`image/svg+xml`, `image/png`) are
-    /// also deferred to Phase 2.
+    /// `dcel_json` is the serialized subgraph (serde_json of [`crate::dcel2::Dcel`]).
+    /// `svg_xml` is an SVG rendering of the same geometry for cross-app paste.
     VectorGeometry {
-        /// Opaque DCEL subgraph bytes (format TBD, Phase 2).
-        data: Vec<u8>,
+        /// JSON-serialized DCEL subgraph.
+        dcel_json: String,
+        /// SVG representation for cross-app paste (e.g. into Inkscape).
+        svg_xml: String,
     },
 
     /// MIDI notes from the piano roll.
@@ -225,11 +225,14 @@ impl ClipboardContent {
             }
 
             // ── VectorGeometry ──────────────────────────────────────────────
-            // TODO (Phase 2): remap DCEL vertex/edge UUIDs once DCEL serialization
-            // is defined.
-            ClipboardContent::VectorGeometry { data } => {
-                (ClipboardContent::VectorGeometry { data: data.clone() }, id_map)
-            }
+            // DCEL uses integer indices (not UUIDs), so no remapping is needed.
+            ClipboardContent::VectorGeometry { dcel_json, svg_xml } => (
+                ClipboardContent::VectorGeometry {
+                    dcel_json: dcel_json.clone(),
+                    svg_xml: svg_xml.clone(),
+                },
+                id_map,
+            ),
 
             // ── MidiNotes ───────────────────────────────────────────────────
             ClipboardContent::MidiNotes { notes } => {
@@ -540,7 +543,8 @@ impl ClipboardManager {
     pub fn copy(&mut self, content: ClipboardContent) {
         let json = serde_json::to_string(&content).unwrap_or_default();
 
-        // Build platform entries (custom MIME always present; PNG secondary for raster).
+        // Build platform entries (custom MIME always present; secondary formats for
+        // specific content types: PNG for raster, SVG for vector geometry).
         let mut entries: Vec<(&str, Vec<u8>)> =
             vec![(LIGHTNINGBEAM_MIME, json.as_bytes().to_vec())];
         if let ClipboardContent::RasterPixels { pixels, width, height } = &content {
@@ -548,6 +552,9 @@ impl ClipboardManager {
                 entries.push(("image/png", png));
             }
         }
+        if let ClipboardContent::VectorGeometry { svg_xml, .. } = &content {
+            entries.push(("image/svg+xml", svg_xml.as_bytes().to_vec()));
+        }
 
         clipboard_platform::set(
             &entries.iter().map(|(m, d)| (*m, d.as_slice())).collect::<Vec<_>>(),

lightningbeam-ui/lightningbeam-core/src/dcel2/import.rs

@@ -0,0 +1,280 @@
+//! DCEL import/extract: merge a serialized DCEL subgraph into a live DCEL,
+//! or extract a subgraph from selected edges for clipboard copy.
+//!
+//! Used by paste to insert clipboard geometry into the current layer,
+//! and by copy to extract a sub-DCEL from the select-tool selection.
+
+use super::{Dcel, EdgeId, FaceId};
+use crate::shape::FillRule;
+use kurbo::{BezPath, CubicBez, Point, Shape, Vec2};
+use std::collections::HashSet;
+
+impl Dcel {
+    /// Import all non-deleted geometry from `source` into `self` using proper
+    /// topological integration.
+    ///
+    /// **Phase 1** — build a closed `BezPath` (shifted by `offset`) for every
+    /// filled face in `source`.
+    ///
+    /// **Phase 2** — insert each clipboard edge via `insert_stroke`, which
+    /// handles intersections with existing edges, vertex snapping, and all
+    /// topological invariants.
+    ///
+    /// **Phase 3** — apply fill colours.  Rather than computing an interior
+    /// point for each live DCEL face (which fails for concave faces), we sample
+    /// a dense grid of points *inside* each clipboard `BezPath` and call
+    /// `find_face_at_point` on each.  Every live face hit by at least one
+    /// sample point gets the clipboard fill colour.  This correctly handles
+    /// the case where one clipboard face is split into several sub-faces at
+    /// intersection points.
+    ///
+    /// `offset` is a translation applied to all positions (`Vec2::ZERO` for an
+    /// exact in-place copy).
+    pub fn import_from(&mut self, source: &Dcel, offset: Vec2) {
+        // ── Phase 1: clipboard face → offset BezPath ─────────────────────────
+        let mut fill_targets: Vec<(BezPath, super::ShapeColor, FillRule)> = Vec::new();
+        for (face_idx, face) in source.faces.iter().enumerate() {
+            if face.deleted || face_idx == 0 {
+                continue;
+            }
+            let Some(color) = face.fill_color else { continue };
+            if let Some(path) =
+                clipboard_face_to_bezpath(source, FaceId(face_idx as u32), offset)
+            {
+                fill_targets.push((path, color, face.fill_rule));
+            }
+        }
+
+        // ── Phase 2: insert each clipboard edge as a topologically-integrated stroke ──
+        // Record the face count before insertion so Phase 3 can distinguish old
+        // faces (keep their colour) from new faces (receive clipboard colour).
+        let faces_count_before = self.faces.len();
+
+        for edge in &source.edges {
+            if edge.deleted {
+                continue;
+            }
+            let c = edge.curve;
+            let shifted = CubicBez::new(
+                Point::new(c.p0.x + offset.x, c.p0.y + offset.y),
+                Point::new(c.p1.x + offset.x, c.p1.y + offset.y),
+                Point::new(c.p2.x + offset.x, c.p2.y + offset.y),
+                Point::new(c.p3.x + offset.x, c.p3.y + offset.y),
+            );
+            self.insert_stroke(
+                &[shifted],
+                edge.stroke_style.clone(),
+                edge.stroke_color,
+                super::DEFAULT_SNAP_EPSILON,
+            );
+        }
+
+        // ── Phase 3: grid-sample each clipboard BezPath, paint hit live faces ─
+        //
+        // For each clipboard face boundary (already offset), lay an N×N grid
+        // over its bounding box.  Every grid point inside the path (non-zero
+        // winding) is handed to `find_face_at_point`.
+        //
+        // Two cases for the returned face:
+        //
+        //  a) Non-F0 face created during Phase 2 (new sub-face of an existing
+        //     face that got split by a clipboard edge).  These have face index
+        //     >= `faces_count_before` and receive the clipboard colour.
+        //
+        //  b) Still in F0 (unbounded face) — this happens when the clipboard
+        //     region extends *outside* all pre-existing faces.  The topology
+        //     code does not auto-create faces inside F0.  In this case we use
+        //     the `cycle_he` returned by `find_face_at_point` to explicitly
+        //     claim that finite F0 cycle as a new face, then colour it.
+        //
+        // Old faces (index < `faces_count_before`) are left untouched so that
+        // pre-existing geometry keeps its original colour.
+        for (fill_path, color, fill_rule) in &fill_targets {
+            let bbox = fill_path.bounding_box();
+            if bbox.width() < 1e-9 || bbox.height() < 1e-9 {
+                continue;
+            }
+
+            let mut painted: HashSet<u32> = HashSet::new();
+
+            // N×N interior grid (avoid exact boundary lines).
+            const N: usize = 8;
+            for iy in 1..=N {
+                for ix in 1..=N {
+                    let x = bbox.min_x()
+                        + (ix as f64 / (N + 1) as f64) * bbox.width();
+                    let y = bbox.min_y()
+                        + (iy as f64 / (N + 1) as f64) * bbox.height();
+                    let pt = Point::new(x, y);
+
+                    // Only consider points confirmed inside the clipboard face.
+                    if fill_path.winding(pt) == 0 {
+                        continue;
+                    }
+
+                    let fq = self.find_face_at_point(pt);
+                    let mut fid = fq.face;
+
+                    // Case (b): point is in the unbounded face but inside the
+                    // clipboard BezPath.  The pasted edge(s) already form a
+                    // closed cycle in F0 — claim it as a new face now.
+                    if fid.0 == 0 && !fq.cycle_he.is_none() {
+                        let new_face = self.alloc_face();
+                        self.assign_cycle_face(fq.cycle_he, new_face);
+                        self.faces[new_face.idx()].outer_half_edge = fq.cycle_he;
+                        fid = new_face;
+                    }
+
+                    if fid.is_none() || fid.0 == 0 {
+                        continue;
+                    }
+
+                    // Only paint faces that were created during this paste.
+                    // Faces that existed before paste keep their original fill.
+                    if fid.idx() < faces_count_before {
+                        continue;
+                    }
+
+                    if painted.insert(fid.0) {
+                        self.faces[fid.idx()].fill_color = Some(*color);
+                        self.faces[fid.idx()].fill_rule = *fill_rule;
+                    }
+                }
+            }
+        }
+    }
+}
+
+/// Build a closed `BezPath` for the outer boundary of a clipboard face,
+/// with all positions shifted by `offset`.
+///
+/// Half-edges are walked CCW (DCEL convention: face is to the left of each
+/// directed edge).  Returns `None` if the face has no outer boundary.
+fn clipboard_face_to_bezpath(source: &Dcel, face_id: FaceId, offset: Vec2) -> Option<BezPath> {
+    let start_he = source.face(face_id).outer_half_edge;
+    if start_he.is_none() {
+        return None;
+    }
+
+    let mut path = BezPath::new();
+    let mut first = true;
+    let mut he_id = start_he;
+    let limit = source.half_edges.len() + 1;
+
+    for _ in 0..limit {
+        let he = source.half_edge(he_id);
+        if he.deleted {
+            break;
+        }
+        let edge = source.edge(he.edge);
+        let c = edge.curve;
+
+        // Forward half-edge → curve goes p0→p3; backward → reversed.
+        let (p0, p1, p2, p3) = if edge.half_edges[0] == he_id {
+            (c.p0, c.p1, c.p2, c.p3)
+        } else {
+            (c.p3, c.p2, c.p1, c.p0)
+        };
+
+        let shift = |p: Point| Point::new(p.x + offset.x, p.y + offset.y);
+
+        if first {
+            path.move_to(shift(p0));
+            first = false;
+        }
+        path.curve_to(shift(p1), shift(p2), shift(p3));
+
+        he_id = he.next;
+        if he_id == start_he {
+            break;
+        }
+    }
+
+    if first {
+        return None;
+    }
+    path.close_path();
+    Some(path)
+}
+
+// ── Extract faces for select-tool copy ───────────────────────────────────────
+
+/// Extract a sub-DCEL containing the faces adjacent to the given edges.
+///
+/// Includes all selected edges, both adjacent faces, and all boundary edges
+/// of those faces.  Used by copy when the select tool has selected strokes.
+pub fn extract_faces_for_edges(dcel: &Dcel, edge_ids: &HashSet<EdgeId>) -> Dcel {
+    if edge_ids.is_empty() {
+        return Dcel::new();
+    }
+
+    let mut face_set: HashSet<u32> = HashSet::new();
+    for &eid in edge_ids {
+        if eid.is_none() || dcel.edge(eid).deleted {
+            continue;
+        }
+        let [he_fwd, he_bwd] = dcel.edge(eid).half_edges;
+        for he_id in [he_fwd, he_bwd] {
+            if !he_id.is_none() {
+                let face = dcel.half_edge(he_id).face;
+                if !face.is_none() && face.0 != 0 {
+                    face_set.insert(face.0);
+                }
+            }
+        }
+    }
+
+    if face_set.is_empty() {
+        return Dcel::new();
+    }
+
+    let mut boundary_edge_ids: HashSet<u32> = HashSet::new();
+    for (i, edge) in dcel.edges.iter().enumerate() {
+        if edge.deleted {
+            continue;
+        }
+        let [he_fwd, he_bwd] = edge.half_edges;
+        let face_fwd =
+            if !he_fwd.is_none() { dcel.half_edge(he_fwd).face } else { FaceId::NONE };
+        let face_bwd =
+            if !he_bwd.is_none() { dcel.half_edge(he_bwd).face } else { FaceId::NONE };
+        if (!face_fwd.is_none() && face_set.contains(&face_fwd.0))
+            || (!face_bwd.is_none() && face_set.contains(&face_bwd.0))
+        {
+            boundary_edge_ids.insert(i as u32);
+        }
+    }
+
+    let mut extracted = dcel.clone();
+
+    let to_remove: Vec<EdgeId> = extracted
+        .edges
+        .iter()
+        .enumerate()
+        .filter_map(|(i, e)| {
+            if !e.deleted && !boundary_edge_ids.contains(&(i as u32)) {
+                Some(EdgeId(i as u32))
+            } else {
+                None
+            }
+        })
+        .collect();
+
+    for eid in to_remove {
+        if !extracted.edges[eid.idx()].deleted {
+            extracted.remove_edge(eid);
+        }
+    }
+
+    for (i, face) in extracted.faces.iter_mut().enumerate() {
+        if i == 0 || face.deleted {
+            continue;
+        }
+        if !face_set.contains(&(i as u32)) {
+            face.fill_color = None;
+            face.image_fill = None;
+        }
+    }
+
+    extracted
+}

lightningbeam-ui/lightningbeam-core/src/dcel2/mod.rs

@@ -11,6 +11,9 @@ pub mod topology;
 pub mod query;
 pub mod stroke;
 pub mod region;
+pub mod import;
+
+pub use import::extract_faces_for_edges;
 
 use crate::shape::{FillRule, ShapeColor, StrokeStyle};
 use kurbo::{CubicBez, Point};

lightningbeam-ui/lightningbeam-core/src/dcel2/stroke.rs

@@ -8,7 +8,7 @@
 //! different intersection positions for the same crossing.
 
 use super::{
-    subsegment_cubic, Dcel, EdgeId, FaceId, HalfEdgeId, VertexId,
+    subsegment_cubic, Dcel, EdgeId, FaceId, VertexId,
 };
 use crate::curve_intersections::find_curve_intersections;
 use crate::shape::{ShapeColor, StrokeStyle};
@@ -379,6 +379,13 @@ impl Dcel {
         }
 
         // 2. Check against all other edges
+        //
+        // Collect (seg_t_on_edge_id, vertex, point, other_tail) for each
+        // intersection so that we can also split edge_id after processing all
+        // other edges.  `other_tail` is the sub-edge of the other edge that
+        // starts at `vertex` (going forward) — used for sector-based face repair.
+        let mut edge_id_splits: Vec<(f64, VertexId, Point, EdgeId)> = Vec::new();
+
         let edge_count = self.edges.len();
         for other_idx in 0..edge_count {
             if self.edges[other_idx].deleted {
@@ -417,11 +424,11 @@ impl Dcel {
                 continue;
             }
 
-            // Sort by edge_t descending — split from end first
+            // Sort by edge_t descending — split other_id from end first
             hits.sort_by(|a, b| b.1.partial_cmp(&a.1).unwrap());
 
             let mut head_end = 1.0_f64;
-            for (_seg_t, original_edge_t, point) in hits {
+            for (seg_t, original_edge_t, point) in hits {
                 let remapped_t = (original_edge_t / head_end)
                     .clamp(ENDPOINT_T_MARGIN, 1.0 - ENDPOINT_T_MARGIN);
 
@@ -431,10 +438,65 @@ impl Dcel {
                 self.snap_edge_endpoints_to_vertex(new_edge, vertex);
 
                 created.push((vertex, new_edge));
+                // Record this intersection for splitting edge_id below.
+                // `new_edge` is other_tail: the sub-edge of other_id going
+                // forward from vertex, used for sector-based face repair.
+                edge_id_splits.push((seg_t, vertex, point, new_edge));
                 head_end = original_edge_t;
             }
         }
 
+        // 3. Split edge_id at every intersection point found above.
+        //
+        // We reuse the vertices already created for the other-edge splits so
+        // the two sides of each crossing share exactly one vertex.
+        //
+        // After all splits, each crossing vertex has 4 outgoing half-edges
+        // whose angular ordering was not maintained by the mechanical splices.
+        // Call rebuild_vertex_fan + repair_face_cycles_at_vertex to fix this
+        // (same pattern as the self-intersection case above).
+        if !edge_id_splits.is_empty() {
+            // Sort by seg_t descending — split edge_id from end first so
+            // edge_id always remains the head piece [0 .. current_split_t].
+            edge_id_splits.sort_by(|a, b| b.0.partial_cmp(&a.0).unwrap());
+
+            // Collect (vertex, editing_tail, other_tail) for sector-based repair.
+            // editing_tail = sub-edge of edge_id going forward from vertex.
+            // other_tail   = sub-edge of the other edge going forward from vertex.
+            let mut touched_info: Vec<(VertexId, EdgeId, EdgeId)> = Vec::new();
+
+            let mut head_end = 1.0_f64;
+            for (original_seg_t, vertex, point, other_tail) in edge_id_splits {
+                if self.edges[edge_id.idx()].deleted {
+                    break;
+                }
+                let remapped_t = (original_seg_t / head_end)
+                    .clamp(ENDPOINT_T_MARGIN, 1.0 - ENDPOINT_T_MARGIN);
+
+                let (_, editing_tail) = self.split_edge_at_vertex(edge_id, remapped_t, vertex);
+                // Snap both pieces to the exact intersection point.
+                self.vertices[vertex.idx()].position = point;
+                self.snap_edge_endpoints_to_vertex(edge_id, vertex);
+                self.snap_edge_endpoints_to_vertex(editing_tail, vertex);
+
+                created.push((vertex, editing_tail));
+                // Only the first crossing for each vertex is repaired; extra
+                // crossings at the same vertex are uncommon and fall back to
+                // the basic fan rebuild inside repair_crossing_vertex.
+                if !touched_info.iter().any(|&(v, _, _)| v == vertex) {
+                    touched_info.push((vertex, editing_tail, other_tail));
+                }
+                head_end = original_seg_t;
+            }
+
+            // Use sector-based face assignment: for each crossing vertex,
+            // rebuild the angular fan and assign faces using the tangent
+            // cross-product rule (editing face wins the overlap region).
+            for (v, editing_tail, other_tail) in touched_info {
+                self.repair_crossing_vertex(v, editing_tail, other_tail);
+            }
+        }
+
         created
     }
 

lightningbeam-ui/lightningbeam-core/src/dcel2/topology.rs

@@ -431,7 +431,7 @@ impl Dcel {
     }
 
     /// Set the face of every half-edge in the cycle starting at `start`.
-    fn assign_cycle_face(&mut self, start: HalfEdgeId, face: FaceId) {
+    pub(crate) fn assign_cycle_face(&mut self, start: HalfEdgeId, face: FaceId) {
         self.half_edges[start.idx()].face = face;
         let mut cur = self.half_edges[start.idx()].next;
         let mut steps = 0;
@@ -908,6 +908,106 @@ impl Dcel {
         new_faces
     }
 
+    /// Repair face assignments at a vertex where `editing_tail` (a sub-edge of
+    /// the edge being dragged) and `other_tail` (a sub-edge of the other edge)
+    /// meet after a crossing split.
+    ///
+    /// The face to the LEFT of each tail's forward direction is used to classify
+    /// sectors at `vertex`. "Editing face wins" on overlap: sectors to the left
+    /// of the editing edge's tangent get `face_a`; remaining sectors to the left
+    /// of the other edge's tangent get `face_b`; all others are exterior (F0).
+    ///
+    /// Steps:
+    /// 1. Read face/tangent for each tail's forward half-edge (before they change).
+    /// 2. `rebuild_vertex_fan` — sort all outgoing half-edges CCW.
+    /// 3. For each angular sector, compute the face from tangent cross products.
+    /// 4. `assign_cycle_face(twin(outgoing[i]), sector_face[i])` for each sector.
+    /// 5. If a face appears in two sectors, the second sector gets a new sub-face
+    ///    with the same fill colour.
+    pub fn repair_crossing_vertex(
+        &mut self,
+        vertex: VertexId,
+        editing_tail: EdgeId,
+        other_tail: EdgeId,
+    ) {
+        use std::collections::HashMap;
+
+        // Read face and tangent direction BEFORE any relinking.
+        // editing_tail.half_edges[0] is the forward HE starting at vertex;
+        // face_a = face to the LEFT of the editing edge's forward direction.
+        let he_a = self.edges[editing_tail.idx()].half_edges[0];
+        let face_a = self.half_edges[he_a.idx()].face;
+        let angle_a = self.outgoing_angle(he_a);
+        let ta = (angle_a.cos(), angle_a.sin());
+
+        let he_b = self.edges[other_tail.idx()].half_edges[0];
+        let face_b = self.half_edges[he_b.idx()].face;
+        let angle_b = self.outgoing_angle(he_b);
+        let tb = (angle_b.cos(), angle_b.sin());
+
+        // Sort outgoing half-edges CCW.
+        self.rebuild_vertex_fan(vertex);
+
+        let outgoing = self.vertex_outgoing(vertex);
+        let n = outgoing.len();
+        if n < 2 {
+            return;
+        }
+
+        // Assign face to each angular sector.
+        // Sector i lies between outgoing[i] and outgoing[(i+1)%n] (CCW).
+        // After rebuild_vertex_fan, twin(outgoing[i]).next = outgoing[(i+1)%n],
+        // so the cycle starting at twin(outgoing[i]) borders sector i.
+        let mut face_first_cycle: HashMap<u32, HalfEdgeId> = HashMap::new();
+
+        for i in 0..n {
+            let he_i = outgoing[i];
+            let he_next = outgoing[(i + 1) % n];
+
+            // Midpoint direction of the CCW angular gap between outgoing[i] and outgoing[i+1].
+            let ai = self.outgoing_angle(he_i);
+            let an = self.outgoing_angle(he_next);
+            let mut delta = an - ai;
+            if delta <= 0.0 {
+                delta += 2.0 * std::f64::consts::PI;
+            }
+            let mid_angle = ai + delta / 2.0;
+            let d = (mid_angle.cos(), mid_angle.sin());
+
+            // cross(t, d) > 0  ⟺  d is to the LEFT of t  ⟺  inside that edge's face.
+            let cross_a = ta.0 * d.1 - ta.1 * d.0;
+            let cross_b = tb.0 * d.1 - tb.1 * d.0;
+
+            let sector_face = if cross_a > 0.0 {
+                face_a // editing face wins; also covers the overlap region
+            } else if cross_b > 0.0 {
+                face_b
+            } else {
+                FaceId(0) // exterior
+            };
+
+            let twin_i = self.half_edges[he_i.idx()].twin;
+
+            if sector_face.0 == 0 {
+                self.assign_cycle_face(twin_i, FaceId(0));
+            } else if face_first_cycle.contains_key(&sector_face.0) {
+                // Second occurrence: create a new sub-face with the same fill.
+                let nf = self.alloc_face();
+                let src = sector_face;
+                self.faces[nf.idx()].fill_color = self.faces[src.idx()].fill_color;
+                self.faces[nf.idx()].image_fill = self.faces[src.idx()].image_fill;
+                self.faces[nf.idx()].fill_rule = self.faces[src.idx()].fill_rule;
+                self.faces[nf.idx()].outer_half_edge = twin_i;
+                self.assign_cycle_face(twin_i, nf);
+            } else {
+                face_first_cycle.insert(sector_face.0, twin_i);
+                self.assign_cycle_face(twin_i, sector_face);
+                // Keep outer_half_edge valid for the original face.
+                self.faces[sector_face.idx()].outer_half_edge = twin_i;
+            }
+        }
+    }
+
     /// Merge vertex `v_remove` into `v_keep`. Both must be at the same position
     /// (or close enough). All half-edges originating from `v_remove` are re-homed
     /// to `v_keep`, and the combined fan is re-sorted by angle.

lightningbeam-ui/lightningbeam-core/src/lib.rs

@@ -47,6 +47,7 @@ pub(crate) mod clipboard_platform;
 pub mod region_select;
 pub mod dcel2;
 pub use dcel2 as dcel;
+pub mod svg_export;
 pub mod snap;
 pub mod webcam;
 pub mod raster_layer;

lightningbeam-ui/lightningbeam-core/src/selection.rs

@@ -159,6 +159,14 @@ pub struct Selection {
     /// Transient UI state — not persisted.
     #[serde(skip)]
     pub raster_floating: Option<RasterFloatingSelection>,
+
+    /// Standalone DCEL subgraph ready for clipboard operations.
+    ///
+    /// Set when a region selection is committed (contains the extracted geometry).
+    /// Cleared when the selection is cleared. Used by clipboard_copy_selection
+    /// to avoid re-extracting the geometry from the live DCEL.
+    #[serde(skip)]
+    pub vector_subgraph: Option<Dcel>,
 }
 
 impl Selection {
@@ -171,6 +179,7 @@ impl Selection {
             selected_clip_instances: Vec::new(),
             raster_selection: None,
             raster_floating: None,
+            vector_subgraph: None,
         }
     }
 
@@ -313,6 +322,7 @@ impl Selection {
         self.selected_vertices.clear();
         self.selected_edges.clear();
         self.selected_faces.clear();
+        self.vector_subgraph = None;
     }
 
     /// Check if any DCEL elements are selected.
@@ -406,6 +416,7 @@ impl Selection {
         self.selected_clip_instances.clear();
         self.raster_selection = None;
         self.raster_floating = None;
+        self.vector_subgraph = None;
     }
 
     /// Check if selection is empty

lightningbeam-ui/lightningbeam-core/src/svg_export.rs

@@ -0,0 +1,220 @@
+//! SVG export from a DCEL subgraph.
+//!
+//! Generates a minimal SVG string containing one `<path>` per filled face,
+//! plus stroked edges. Used as the secondary clipboard format for cross-app paste.
+
+use crate::dcel2::{Dcel, FaceId, HalfEdgeId};
+use kurbo::CubicBez;
+
+/// Convert a DCEL to an SVG string.
+///
+/// Each non-unbounded face with a fill color becomes a `<path fill="..."/>`.
+/// Each edge with a stroke becomes a `<path stroke="..."/>`.
+/// Coordinates are document-space (no transform applied).
+pub fn dcel_to_svg(dcel: &Dcel) -> String {
+    // Compute bounding box from vertex positions.
+    let mut min_x = f64::MAX;
+    let mut min_y = f64::MAX;
+    let mut max_x = f64::MIN;
+    let mut max_y = f64::MIN;
+
+    for v in &dcel.vertices {
+        if !v.deleted {
+            min_x = min_x.min(v.position.x);
+            min_y = min_y.min(v.position.y);
+            max_x = max_x.max(v.position.x);
+            max_y = max_y.max(v.position.y);
+        }
+    }
+
+    if min_x == f64::MAX {
+        return r#"<svg xmlns="http://www.w3.org/2000/svg"/>"#.to_string();
+    }
+
+    // Add a small margin.
+    let margin = 2.0;
+    let vx = min_x - margin;
+    let vy = min_y - margin;
+    let vw = (max_x - min_x) + margin * 2.0;
+    let vh = (max_y - min_y) + margin * 2.0;
+
+    let mut svg = format!(
+        r#"<svg xmlns="http://www.w3.org/2000/svg" viewBox="{vx:.3} {vy:.3} {vw:.3} {vh:.3}">"#
+    );
+
+    // Emit filled faces.
+    for (face_idx, face) in dcel.faces.iter().enumerate() {
+        if face.deleted || face_idx == 0 {
+            continue;
+        }
+        let fill_color = match &face.fill_color {
+            Some(c) => format!("rgba({},{},{},{})", c.r, c.g, c.b, c.a as f32 / 255.0),
+            None => continue,
+        };
+
+        let face_id = FaceId(face_idx as u32);
+        let path_d = face_boundary_to_svg_path(dcel, face_id);
+        if path_d.is_empty() {
+            continue;
+        }
+
+        svg.push_str(&format!(r#"<path fill="{fill_color}" d="{path_d}"/>"#));
+    }
+
+    // Emit stroked edges.
+    for edge in &dcel.edges {
+        if edge.deleted {
+            continue;
+        }
+        let (stroke_color, stroke_width) = match (&edge.stroke_color, &edge.stroke_style) {
+            (Some(c), Some(s)) => (
+                format!("rgba({},{},{},{})", c.r, c.g, c.b, c.a as f32 / 255.0),
+                s.width,
+            ),
+            _ => continue,
+        };
+
+        let path_d = cubic_to_svg_path(&edge.curve);
+        svg.push_str(&format!(
+            r#"<path fill="none" stroke="{stroke_color}" stroke-width="{stroke_width:.3}" d="{path_d}"/>"#
+        ));
+    }
+
+    svg.push_str("</svg>");
+    svg
+}
+
+/// Walk a face's outer boundary half-edges and build an SVG path string.
+fn face_boundary_to_svg_path(dcel: &Dcel, face_id: FaceId) -> String {
+    let face = dcel.face(face_id);
+    let start_he = face.outer_half_edge;
+    if start_he.is_none() {
+        return String::new();
+    }
+
+    let mut path = String::new();
+    let mut first = true;
+    let mut he_id = start_he;
+
+    // Safety limit to prevent infinite loops on malformed DCELs.
+    let limit = dcel.half_edges.len() + 1;
+    let mut count = 0;
+
+    loop {
+        if count > limit {
+            break;
+        }
+        count += 1;
+
+        let he = dcel.half_edge(he_id);
+        if he.deleted {
+            break;
+        }
+
+        let edge = dcel.edge(he.edge);
+        // Determine curve direction: forward half-edge is half_edges[0].
+        let curve = if edge.half_edges[0] == he_id {
+            edge.curve
+        } else {
+            // Reverse the cubic bezier.
+            let c = edge.curve;
+            CubicBez::new(c.p3, c.p2, c.p1, c.p0)
+        };
+
+        if first {
+            path.push_str(&format!("M {:.3} {:.3} ", curve.p0.x, curve.p0.y));
+            first = false;
+        }
+
+        path.push_str(&format!(
+            "C {:.3} {:.3} {:.3} {:.3} {:.3} {:.3} ",
+            curve.p1.x, curve.p1.y,
+            curve.p2.x, curve.p2.y,
+            curve.p3.x, curve.p3.y,
+        ));
+
+        he_id = he.next;
+        if he_id == start_he {
+            break;
+        }
+    }
+
+    if !path.is_empty() {
+        path.push('Z');
+    }
+
+    // Also handle inner boundaries (holes).
+    for &inner_he_start in &face.inner_half_edges {
+        if inner_he_start.is_none() {
+            continue;
+        }
+        let inner = inner_boundary_to_svg_path(dcel, inner_he_start);
+        if !inner.is_empty() {
+            path.push(' ');
+            path.push_str(&inner);
+        }
+    }
+
+    path
+}
+
+fn inner_boundary_to_svg_path(dcel: &Dcel, start_he: HalfEdgeId) -> String {
+    let mut path = String::new();
+    let mut first = true;
+    let mut he_id = start_he;
+    let limit = dcel.half_edges.len() + 1;
+    let mut count = 0;
+
+    loop {
+        if count > limit {
+            break;
+        }
+        count += 1;
+
+        let he = dcel.half_edge(he_id);
+        if he.deleted {
+            break;
+        }
+
+        let edge = dcel.edge(he.edge);
+        let curve = if edge.half_edges[0] == he_id {
+            edge.curve
+        } else {
+            let c = edge.curve;
+            CubicBez::new(c.p3, c.p2, c.p1, c.p0)
+        };
+
+        if first {
+            path.push_str(&format!("M {:.3} {:.3} ", curve.p0.x, curve.p0.y));
+            first = false;
+        }
+
+        path.push_str(&format!(
+            "C {:.3} {:.3} {:.3} {:.3} {:.3} {:.3} ",
+            curve.p1.x, curve.p1.y,
+            curve.p2.x, curve.p2.y,
+            curve.p3.x, curve.p3.y,
+        ));
+
+        he_id = he.next;
+        if he_id == start_he {
+            break;
+        }
+    }
+
+    if !path.is_empty() {
+        path.push('Z');
+    }
+    path
+}
+
+/// Convert a single cubic bezier to an SVG path string.
+fn cubic_to_svg_path(curve: &CubicBez) -> String {
+    format!(
+        "M {:.3} {:.3} C {:.3} {:.3} {:.3} {:.3} {:.3} {:.3}",
+        curve.p0.x, curve.p0.y,
+        curve.p1.x, curve.p1.y,
+        curve.p2.x, curve.p2.y,
+        curve.p3.x, curve.p3.y,
+    )
+}

lightningbeam-ui/lightningbeam-editor/src/main.rs

@@ -2123,7 +2123,36 @@ impl EditorApp {
 
             self.clipboard_manager.copy(content);
         } else if self.selection.has_dcel_selection() {
-            // TODO: DCEL copy/paste deferred to Phase 2 (serialize subgraph)
+            let subgraph = if let Some(dcel) = self.selection.vector_subgraph.take() {
+                // Region selection: the sub-DCEL was pre-extracted on commit.
+                dcel
+            } else {
+                // Select tool: extract faces adjacent to the selected edges from the live DCEL.
+                let active_layer_id = match self.active_layer_id {
+                    Some(id) => id,
+                    None => return,
+                };
+                let document = self.action_executor.document();
+                let Some(lightningbeam_core::layer::AnyLayer::Vector(vl)) = document.get_layer(&active_layer_id) else {
+                    return;
+                };
+                let Some(live_dcel) = vl.dcel_at_time(self.playback_time) else {
+                    return;
+                };
+                let selected_edges = self.selection.selected_edges().clone();
+                lightningbeam_core::dcel2::extract_faces_for_edges(live_dcel, &selected_edges)
+            };
+
+            let dcel_json = serde_json::to_string(&subgraph).unwrap_or_default();
+            let svg_xml = lightningbeam_core::svg_export::dcel_to_svg(&subgraph);
+            self.clipboard_manager.copy(
+                lightningbeam_core::clipboard::ClipboardContent::VectorGeometry {
+                    dcel_json,
+                    svg_xml,
+                },
+            );
+            // Restore the subgraph so a subsequent cut can also delete.
+            self.selection.vector_subgraph = Some(subgraph);
         }
     }
 
@@ -2223,7 +2252,40 @@ impl EditorApp {
                 None => return,
             };
 
-            // Delete selected edges via snapshot-based ModifyDcelAction
+            // Region selection case: faces are selected but no edges.
+            // The inside geometry was already extracted from the live DCEL;
+            // commit the current state (outside + boundary) using the
+            // pre-boundary snapshot as the "before" for undo.
+            if self.selection.selected_edges().is_empty() {
+                if let Some(region_sel) = self.region_selection.take() {
+                    // dcel_snapshot = state before boundary was inserted.
+                    // Current document DCEL = outside portion only (boundary edges present).
+                    // We commit the snapshot as "before" and the current state as "after",
+                    // then drop the region selection so it is not merged back.
+                    let document = self.action_executor.document();
+                    if let Some(lightningbeam_core::layer::AnyLayer::Vector(vl)) =
+                        document.get_layer(&region_sel.layer_id)
+                    {
+                        if let Some(dcel_after) = vl.dcel_at_time(region_sel.time) {
+                            let action = lightningbeam_core::actions::ModifyDcelAction::new(
+                                region_sel.layer_id,
+                                region_sel.time,
+                                region_sel.dcel_snapshot.clone(),
+                                dcel_after.clone(),
+                                "Cut/delete region selection",
+                            );
+                            if let Err(e) = self.action_executor.execute(Box::new(action)) {
+                                eprintln!("Delete region selection failed: {}", e);
+                            }
+                        }
+                    }
+                    // region_sel is dropped; the stage pane will see region_selection == None.
+                }
+                self.selection.clear_dcel_selection();
+                return;
+            }
+
+            // Select-tool case: delete the selected edges.
             let edge_ids: Vec<lightningbeam_core::dcel::EdgeId> =
                 self.selection.selected_edges().iter().copied().collect();
 
@@ -2378,8 +2440,42 @@ impl EditorApp {
                     self.selection.add_clip_instance(id);
                 }
             }
-            ClipboardContent::VectorGeometry { .. } => {
-                // TODO (Phase 2): paste DCEL subgraph once vector serialization is defined.
+            ClipboardContent::VectorGeometry { dcel_json, .. } => {
+                // Deserialize the subgraph and merge it into the live DCEL.
+                let clipboard_dcel: lightningbeam_core::dcel2::Dcel =
+                    match serde_json::from_str(&dcel_json) {
+                        Ok(d) => d,
+                        Err(e) => {
+                            eprintln!("Paste: failed to deserialize vector geometry: {e}");
+                            return;
+                        }
+                    };
+
+                let active_layer_id = match self.active_layer_id {
+                    Some(id) => id,
+                    None => return,
+                };
+
+                let document = self.action_executor.document();
+                let Some(lightningbeam_core::layer::AnyLayer::Vector(vl)) =
+                    document.get_layer(&active_layer_id) else { return };
+                let Some(dcel_before) = vl.dcel_at_time(self.playback_time) else { return };
+
+                let mut dcel_after = dcel_before.clone();
+                // Paste with a small nudge so it is visually distinct from the original.
+                let nudge = vello::kurbo::Vec2::new(10.0, 10.0);
+                dcel_after.import_from(&clipboard_dcel, nudge);
+
+                let action = lightningbeam_core::actions::ModifyDcelAction::new(
+                    active_layer_id,
+                    self.playback_time,
+                    dcel_before.clone(),
+                    dcel_after,
+                    "Paste vector geometry",
+                );
+                if let Err(e) = self.action_executor.execute(Box::new(action)) {
+                    eprintln!("Paste vector geometry failed: {e}");
+                }
             }
             ClipboardContent::Layers { .. } => {
                 // TODO: insert copied layers as siblings at the current selection point.
@@ -2902,6 +2998,9 @@ impl EditorApp {
                     if let Some((clip_id, notes)) = midi_update {
                         self.rebuild_midi_cache_entry(clip_id, &notes);
                     }
+                    // Stale vertex/edge/face IDs from before the undo would
+                    // crash selection rendering on the restored (smaller) DCEL.
+                    self.selection.clear_dcel_selection();
                 }
             }
             MenuAction::Redo => {
@@ -2938,6 +3037,7 @@ impl EditorApp {
                     if let Some((clip_id, notes)) = midi_update {
                         self.rebuild_midi_cache_entry(clip_id, &notes);
                     }
+                    self.selection.clear_dcel_selection();
                 }
             }
             MenuAction::Cut => {

lightningbeam-ui/lightningbeam-editor/src/panes/stage.rs

@@ -4292,6 +4292,11 @@ impl StagePane {
             }
         }
 
+        // Store the extracted DCEL as the clipboard-ready vector subgraph.
+        // This allows clipboard_copy_selection to serialize it without needing
+        // to re-extract geometry from the live DCEL.
+        shared.selection.vector_subgraph = Some(selected_dcel.clone());
+
         // Store region selection state with extracted DCEL
         *shared.region_selection = Some(lightningbeam_core::selection::RegionSelection {
             region_path,