diff --git a/Cargo.toml b/Cargo.toml
index edc9c9b..ece5f62 100644
--- a/Cargo.toml
+++ b/Cargo.toml
@@ -70,6 +70,7 @@ features = ["rstar"]
 
 [dev-dependencies]
 serde_json.workspace = true
+proptest = "1.6"
 
 [package.metadata.docs.rs]
 cargo-args = ["-Zunstable-options", "-Zrustdoc-scrape-examples"]
diff --git a/crates/planar-incr-embed/Cargo.toml b/crates/planar-incr-embed/Cargo.toml
index 12aaaef..0707f23 100644
--- a/crates/planar-incr-embed/Cargo.toml
+++ b/crates/planar-incr-embed/Cargo.toml
@@ -24,10 +24,8 @@ optional = true
 
 [dev-dependencies]
 ron = "0.8"
+serde.workspace = true
 
 [dev-dependencies.insta]
 version = "1.42"
 features = ["json"]
-
-[dev-dependencies.serde]
-workspace = true
diff --git a/proptest-regressions/math/cyclic_search.txt b/proptest-regressions/math/cyclic_search.txt
new file mode 100644
index 0000000..4e48f83
--- /dev/null
+++ b/proptest-regressions/math/cyclic_search.txt
@@ -0,0 +1,11 @@
+# SPDX-FileCopyrightText: 2025 Topola contributors
+#
+# SPDX-License-Identifier: MIT
+
+# Seeds for failure cases proptest has generated in the past. It is
+# automatically read and these particular cases re-run before any
+# novel cases are generated.
+#
+# It is recommended to check this file in to source control so that
+# everyone who runs the test benefits from these saved cases.
+cc 10a8063ed53caab61327919fad368bd440c684021e1022082d6bf2098e19aea9 # shrinks to len = 4, offset = 0, amount_true = 238
diff --git a/src/math/cyclic_search.rs b/src/math/cyclic_search.rs
new file mode 100644
index 0000000..d444121
--- /dev/null
+++ b/src/math/cyclic_search.rs
@@ -0,0 +1,299 @@
+// SPDX-FileCopyrightText: 2025 Topola contributors
+//
+// SPDX-License-Identifier: MIT
+
+use core::{mem::take, ops::Range};
+
+/// generate a breadth-first search list for given bounds and level
+fn breadth4level(bounds: Range<usize>, level: u8, mut callback: impl FnMut(Range<usize>) -> bool) {
+    // level is the exponent of 2
+    let block_length = (bounds.end - bounds.start) >> level;
+    let blocks_count = (bounds.end - bounds.start) / block_length;
+    if blocks_count == 0 {
+        return;
+    }
+    for i in 0..(blocks_count - 1) {
+        let block_start = bounds.start + i * block_length;
+        debug_assert!(block_start + block_length < bounds.end);
+        if !callback(block_start..(block_start + block_length)) {
+            return;
+        }
+    }
+    let block_start = bounds.start + (blocks_count - 1) * block_length;
+    debug_assert!(block_start + block_length <= bounds.end);
+    callback(block_start..bounds.end);
+}
+
+#[derive(Clone, Copy)]
+enum TriState<T> {
+    Nothing,
+    Got(T),
+    Fixed(T),
+}
+
+impl<T> Default for TriState<T> {
+    fn default() -> Self {
+        TriState::Nothing
+    }
+}
+
+impl<T> TriState<T> {
+    fn update(&mut self, value: T) -> bool {
+        match self {
+            TriState::Fixed(_) => false,
+            _ => {
+                *self = TriState::Got(value);
+                true
+            }
+        }
+    }
+
+    fn fix(&mut self) {
+        *self = match take(self) {
+            TriState::Got(x) => TriState::Fixed(x),
+            x => x,
+        };
+    }
+
+    fn is_fixed(&self) -> bool {
+        matches!(self, TriState::Fixed(_))
+    }
+}
+
+impl<T> From<TriState<T>> for Option<T> {
+    fn from(x: TriState<T>) -> Self {
+        match x {
+            TriState::Nothing => None,
+            TriState::Got(x) | TriState::Fixed(x) => Some(x),
+        }
+    }
+}
+
+struct Discover {
+    pos_false: TriState<usize>,
+    pos_true: TriState<usize>,
+}
+
+impl Discover {
+    fn new() -> Self {
+        Self {
+            pos_false: TriState::Nothing,
+            pos_true: TriState::Nothing,
+        }
+    }
+
+    fn update(&mut self, pos: usize, value: bool) {
+        match value {
+            false => {
+                self.pos_false.update(pos);
+                self.pos_true.fix();
+            }
+            true => {
+                self.pos_true.update(pos);
+                self.pos_false.fix();
+            }
+        }
+    }
+
+    fn is_finished_minimal(&self) -> bool {
+        self.pos_false.is_fixed() || self.pos_true.is_fixed()
+    }
+
+    fn is_finished(&self) -> bool {
+        self.pos_false.is_fixed() && self.pos_true.is_fixed()
+    }
+
+    fn results(&self) -> (Option<usize>, Option<usize>) {
+        (self.pos_false.into(), self.pos_true.into())
+    }
+}
+
+/// A brute-force implementation of [`cyclic_breadth_binary_search`].
+fn cbps_brute_force<EF>(
+    bounds: core::ops::Range<usize>,
+    eval: &EF,
+) -> (Option<usize>, Option<usize>)
+where
+    EF: Fn(usize) -> bool,
+{
+    let mut discover = Discover::new();
+
+    for i in bounds {
+        discover.update(i, eval(i));
+        if discover.is_finished() {
+            break;
+        }
+    }
+
+    discover.results()
+}
+
+/// Search for the largest index inside the bounds which still fulfills the condition
+fn exponential_search<T, EF>(
+    eval: &EF,
+    expected_value: T,
+    mut bounds: core::ops::Range<usize>,
+) -> Option<usize>
+where
+    EF: Fn(usize) -> T,
+    T: Eq + core::fmt::Debug,
+{
+    assert!(bounds.start <= bounds.end);
+    if bounds.is_empty() || eval(bounds.start) != expected_value {
+        return None;
+    }
+
+    let mut largest_checked = bounds.start;
+    while (bounds.start + 1) < bounds.end {
+        let len = bounds.end - bounds.start;
+        for level in 0..64u8 {
+            let mut index = 1 << level;
+            if index >= len {
+                break;
+            }
+            index += bounds.start;
+            if eval(index) != expected_value {
+                bounds.end = index;
+                break;
+            }
+            largest_checked = index;
+        }
+
+        bounds.start = largest_checked;
+        // this implies that `bounds.start` doesn't have to get checked again
+    }
+
+    debug_assert_eq!(eval(largest_checked), expected_value);
+    Some(largest_checked)
+}
+
+/// Perform a breadth-first search on an induced binary tree on the list,
+/// searching for the bounds of the partition induced by `eval`,
+/// returning the last item indices in the `false` and `true` blocks
+pub fn cyclic_breadth_partition_search<EF>(
+    bounds: Range<usize>,
+    eval: EF,
+) -> (Option<usize>, Option<usize>)
+where
+    EF: Fn(usize) -> bool,
+{
+    if bounds.is_empty() {
+        return (None, None);
+    }
+
+    // discover gaps (true is a gap for false and vice versa)
+    let mut discover = Discover::new();
+
+    for i in 0..((bounds.end - bounds.start).ilog2() as u8) {
+        breadth4level(bounds.clone(), i, |bounds| {
+            let middle = bounds.start + (bounds.end - bounds.start) / 2;
+            discover.update(middle, eval(middle));
+            !discover.is_finished_minimal()
+        });
+        if discover.is_finished_minimal() {
+            break;
+        }
+    }
+
+    // brute force on failure
+    if !discover.is_finished() {
+        return cbps_brute_force(bounds, &eval);
+    }
+
+    let (pos_false, pos_true) = discover.results();
+    let (mut pos_false, mut pos_true) = (pos_false.unwrap(), pos_true.unwrap());
+
+    // discover bounds
+    debug_assert_ne!(pos_false, pos_true);
+
+    // whatever block is at the beginning has
+    // its end somewhere strictly before the other block
+    // either:
+    // - the later block continues at the beginning
+    //   format: L...!L...L...
+    // - or the later block doesn't continue at the beginning
+    //   format: !L...L...
+    let val_start = eval(bounds.start);
+
+    {
+        let (pos_start, pos_next) = match val_start {
+            false => (&mut pos_false, &mut pos_true),
+            true => (&mut pos_true, &mut pos_false),
+        };
+
+        *pos_start = exponential_search(&eval, val_start, bounds.start..*pos_next).unwrap();
+        *pos_next = exponential_search(&eval, !val_start, *pos_start + 1..bounds.end).unwrap();
+    }
+
+    (Some(pos_false), Some(pos_true))
+}
+
+#[cfg(test)]
+mod tests {
+    use super::{cbps_brute_force, cyclic_breadth_partition_search as cbps};
+    use proptest::prelude::*;
+
+    fn cbps_assert_eq<T, PF>(list: &[T], partition: PF) -> (Option<usize>, Option<usize>)
+    where
+        PF: Fn(&T) -> bool,
+        T: Eq + core::fmt::Debug,
+    {
+        let eval = &|i: usize| partition(&list[i]);
+        let res_expected = cbps_brute_force(0..list.len(), eval);
+        assert_eq!(cbps(0..list.len(), eval), res_expected);
+        res_expected
+    }
+
+    #[test]
+    fn cbps_bpw3_simple00() {
+        let list = &[false, false, false, true, true];
+        assert_eq!(cbps_assert_eq(list, |i| *i), (Some(2), Some(4)));
+    }
+
+    #[test]
+    fn cbps_bpw3_cont_false() {
+        let list = &[false, false, false];
+        assert_eq!(cbps_assert_eq(list, |i| *i), (Some(2), None));
+    }
+
+    #[test]
+    fn cbps_bpw3_cont_true() {
+        let list = &[true, true];
+        assert_eq!(cbps_assert_eq(list, |i| *i), (None, Some(1)));
+    }
+
+    #[test]
+    fn cbps_bpw3_simple01() {
+        let list = &[true, false, false, false, true, true];
+        assert_eq!(cbps_assert_eq(list, |i| *i), (Some(3), Some(0)));
+    }
+
+    #[test]
+    fn cbps_bpw3_1exception() {
+        let list = &[true, false, true, true, true, true];
+        assert_eq!(cbps_assert_eq(list, |i| *i), (Some(1), Some(0)));
+    }
+
+    proptest! {
+        #![proptest_config(ProptestConfig {
+            cases: 4096, .. ProptestConfig::default()
+        })]
+
+        #[test]
+        fn cbps_arbitrary(len in 1..4096usize, offset in 0..4096usize, amount_true in 0..4096usize) {
+            let offset = offset % len;
+            let amount_true = amount_true % len;
+
+            let mut list = vec![false; len];
+            for i in offset..(offset + amount_true) {
+                let i = i % len;
+                list[i] = true;
+            }
+            let eval = &|i: usize| list[i];
+            prop_assert_eq!(
+                cbps(0..list.len(), eval),
+                cbps_brute_force(0..list.len(), eval)
+            );
+        }
+    }
+}
diff --git a/src/math/mod.rs b/src/math/mod.rs
index cffa601..e8f31c9 100644
--- a/src/math/mod.rs
+++ b/src/math/mod.rs
@@ -6,6 +6,12 @@ use geo::algorithm::line_measures::{Distance, Euclidean};
 use geo::{geometry::Point, point, Line};
 pub use specctra_core::math::{Circle, PointWithRotation};
 
+mod cyclic_search;
+pub use cyclic_search::*;
+
+mod polygon_tangents;
+pub use polygon_tangents::*;
+
 mod tangents;
 pub use tangents::*;
 
diff --git a/src/math/polygon_tangents.rs b/src/math/polygon_tangents.rs
new file mode 100644
index 0000000..4356b12
--- /dev/null
+++ b/src/math/polygon_tangents.rs
@@ -0,0 +1,114 @@
+// SPDX-FileCopyrightText: 2025 Topola contributors
+//
+// SPDX-License-Identifier: MIT
+
+use super::{between_vectors, cyclic_breadth_partition_search};
+use geo::Point;
+
+#[derive(Clone, Debug, thiserror::Error, PartialEq)]
+pub enum PolyTangentException<I> {
+    #[error("trying to target empty polygon")]
+    EmptyTargetPolygon { origin: Point },
+
+    #[error("invalid polygon tangent arguments")]
+    InvalidData {
+        poly_ext: Box<[(Point, I)]>,
+        origin: Point,
+    },
+}
+
+/// Calculates the tangents to the polygon exterior `poly_ext` oriented `cw?=poly_ext_is_cw`
+/// going through point `origin`.
+pub fn poly_ext_tangent_points<I: Copy>(
+    poly_ext: &[(Point, I)],
+    poly_ext_is_cw: bool,
+    origin: Point,
+) -> Result<(I, I), PolyTangentException<I>> {
+    if poly_ext.is_empty() {
+        return Err(PolyTangentException::EmptyTargetPolygon { origin });
+    }
+
+    let (pos_false, pos_true) = cyclic_breadth_partition_search(0..poly_ext.len(), &|i: usize| {
+        let prev = &poly_ext[(poly_ext.len() + i - 1) % poly_ext.len()];
+        let cur = &poly_ext[i];
+        let next = &poly_ext[(i + 1) % poly_ext.len()];
+
+        // local coordinate system with origin at `cur.0`.
+        between_vectors(cur.0 - origin, cur.0 - prev.0, cur.0 - next.0)
+    });
+
+    let (mut pos_false, mut pos_true) =
+        if let (Some(pos_false), Some(pos_true)) = (pos_false, pos_true) {
+            (pos_false, pos_true)
+        } else {
+            return Err(PolyTangentException::InvalidData {
+                poly_ext: poly_ext.to_vec().into_boxed_slice(),
+                origin,
+            });
+        };
+
+    // * `pos_false` points to the maximum
+    // * `pos_true`  points to the minimum
+
+    // NOTE: although pos_{false,true} are vertex indices, they are actually
+    // referring to the "critical" segment(s) (pos_false, pos_false + 1) (and resp. for pos_true).
+    // because that is where the `between_vectors` result flips.
+    // These critical segments are independent of CW/CCW.
+
+    // if `poly_ext` is oriented CCW, then
+    // * `pos_false` will be one too early, and
+    // * `pos_true`  will be correct.
+
+    // if `poly_ext` is oriented CW, then
+    // * `pos_false` will be correct.
+    // * `pos_true`  will be one too early, and
+
+    // TODO: can we (without too much overhead) determine if `poly_ext` is CW or CCW?
+
+    if poly_ext_is_cw {
+        pos_true += 1;
+        pos_true %= poly_ext.len();
+    } else {
+        pos_false += 1;
+        pos_false %= poly_ext.len();
+    }
+
+    Ok((poly_ext[pos_true].1, poly_ext[pos_false].1))
+}
+
+#[cfg(test)]
+mod tests {
+    use super::poly_ext_tangent_points as petp;
+    use crate::drawing::dot::FixedDotIndex;
+    use geo::point;
+
+    #[test]
+    fn petp00() {
+        let poly_ext = &[
+            (point! { x: 0.0, y: 0.0 }, FixedDotIndex::new(0.into())),
+            (point! { x: 1.0, y: 0.0 }, FixedDotIndex::new(1.into())),
+            (point! { x: 1.0, y: 1.0 }, FixedDotIndex::new(2.into())),
+            (point! { x: 0.0, y: 1.0 }, FixedDotIndex::new(3.into())),
+        ];
+        let origin = point! { x: 0.5, y: -1.0 };
+        assert_eq!(
+            petp(poly_ext, false, origin),
+            Ok((FixedDotIndex::new(1.into()), FixedDotIndex::new(0.into())))
+        );
+    }
+
+    #[test]
+    fn petp00cw() {
+        let poly_ext = &[
+            (point! { x: 0.0, y: 0.0 }, FixedDotIndex::new(0.into())),
+            (point! { x: 0.0, y: 1.0 }, FixedDotIndex::new(3.into())),
+            (point! { x: 1.0, y: 1.0 }, FixedDotIndex::new(2.into())),
+            (point! { x: 1.0, y: 0.0 }, FixedDotIndex::new(1.into())),
+        ];
+        let origin = point! { x: 0.5, y: -1.0 };
+        assert_eq!(
+            petp(poly_ext, true, origin),
+            Ok((FixedDotIndex::new(1.into()), FixedDotIndex::new(0.into())))
+        );
+    }
+}