feat: add lambda fractal

src/app.rs

@@ -46,6 +46,7 @@ const KINDS: &[(FractalKind, &str)] = &[
     (FractalKind::Perpendicular, "Perpendicular"),
     (FractalKind::Buffalo, "Buffalo"),
     (FractalKind::Phoenix, "Phoenix"),
+    (FractalKind::Lambda, "Lambda"),
 ];
 
 /// UI label for a fractal kind.
@@ -60,7 +61,7 @@ fn kind_label(kind: FractalKind) -> &'static str {
 type JuliaPreset = (&'static str, f64, f64, u32, Option<(f64, f64)>);
 
 /// Nice-looking Julia constants offered as presets.
-const JULIA_PRESETS: [&[JuliaPreset]; FractalKind::Phoenix as usize + 1] = [
+const JULIA_PRESETS: [&[JuliaPreset]; FractalKind::Lambda as usize + 1] = [
     &[
         ("dendrite", -0.8, 0.156, 400, None),
         ("rabbit", -0.123, 0.745, 400, None),
@@ -78,6 +79,7 @@ const JULIA_PRESETS: [&[JuliaPreset]; FractalKind::Phoenix as usize + 1] = [
         ("archipelago 1", -0.415, -0.267, 500, Some((-0.556, 0.253))),
         ("archipelago 2", -0.556, 0.253, 500, Some((-0.415, -0.267))),
     ],
+    &[],
 ];
 
 type SetPreset = (
@@ -92,7 +94,7 @@ type SetPreset = (
 /// Curated beautiful locations offered as one-click presets.
 /// Each is `(name, center_re, center_im, half_height, iterations)`; the centers
 /// are decimals parsed at full precision so deep places stay sharp.
-const SET_PRESETS: [&[SetPreset]; FractalKind::Phoenix as usize + 1] = [
+const SET_PRESETS: [&[SetPreset]; FractalKind::Lambda as usize + 1] = [
     &[
         (
             "Seahorse Valley",
@@ -142,6 +144,7 @@ const SET_PRESETS: [&[SetPreset]; FractalKind::Phoenix as usize + 1] = [
         1000,
         Some((-0.9, -0.49)),
     )],
+    &[],
 ];
 
 /// Parameters a reference orbit was (or will be) computed for. Used to decide
@@ -153,6 +156,7 @@ struct RequestKey {
     julia: bool,
     julia_c: (f64, f64),
     phoenix_p: (f64, f64),
+    lambda_l: (f64, f64),
     iter: u32,
     kind: FractalKind,
     power: u32,
@@ -196,6 +200,13 @@ struct AnimState {
     phoenix_base: (f64, f64),
     phoenix_angle: f64,
 
+    /// Drift the Lambda distortion `λ` around a circle.
+    lambda: bool,
+    lambda_speed: f32,
+    lambda_radius: f64,
+    lambda_base: (f64, f64),
+    lambda_angle: f64,
+
     /// Continuously zoom toward the current center.
     zoom: bool,
     /// e-folds per second; positive zooms in, negative zooms out.
@@ -217,6 +228,11 @@ impl Default for AnimState {
             phoenix_radius: 0.08,
             phoenix_base: (0.0, 0.0),
             phoenix_angle: 0.0,
+            lambda: false,
+            lambda_speed: 0.05,
+            lambda_radius: 0.08,
+            lambda_base: (0.0, 0.0),
+            lambda_angle: 0.0,
             zoom: false,
             zoom_speed: 0.5,
         }
@@ -234,6 +250,8 @@ pub struct FractalApp {
     julia_c: (f64, f64),
     /// Distortion constant `p` for the Phoenix kind (`z^2 + c + p·z_{n-1}`).
     phoenix_p: (f64, f64),
+    /// Distortion constant `l` for the Lambda kind (`l·z(1 - z_{n-1})`).
+    lambda_l: (f64, f64),
     max_iterations: u32,
     /// When set, `max_iterations` tracks the zoom depth automatically (so deep
     /// zooms stay sharp without hand-tuning); the manual slider takes over when
@@ -356,6 +374,7 @@ impl FractalApp {
             power: 3,
             julia_c: (-0.8, 0.156),
             phoenix_p: (-0.5, 0.0),
+            lambda_l: (-0.5, 0.0),
             max_iterations: 512,
             auto_iterations: true,
             color_scale: 0.15,
@@ -517,6 +536,7 @@ impl FractalApp {
             iterations: self.max_iterations,
             julia_c: self.julia_c,
             phoenix_p: self.phoenix_p,
+            lambda_l: self.lambda_l,
             color_scale: self.color_scale,
             color_offset: self.color_offset,
             palette: self.palette,
@@ -534,6 +554,7 @@ impl FractalApp {
         self.power = s.power.clamp(2, 8);
         self.julia_c = s.julia_c;
         self.phoenix_p = s.phoenix_p;
+        self.lambda_l = s.lambda_l;
         self.color_scale = s.color_scale;
         self.color_offset = s.color_offset;
         self.palette = (s.palette as usize).min(PALETTE_NAMES.len() - 1) as u32;
@@ -582,6 +603,7 @@ impl FractalApp {
             FractalKind::Perpendicular => (-0.5, 0.0, 1.5),
             FractalKind::Buffalo => (-0.5, -0.5, 1.5),
             FractalKind::Phoenix => (0.0, 0.0, 1.6),
+            FractalKind::Lambda => (0.0, 0.0, 1.6),
         };
         ViewState::with_center(big_from_f64(cr, 53), big_from_f64(ci, 53), hh)
     }
@@ -594,6 +616,7 @@ impl FractalApp {
             julia: matches!(self.mode, FractalMode::Julia),
             julia_c: self.julia_c,
             phoenix_p: self.phoenix_p,
+            lambda_l: self.lambda_l,
             iter: self.max_iterations,
             kind: self.kind,
             power: self.power,
@@ -609,7 +632,8 @@ impl FractalApp {
 
     /// Whether the reference should be (re)computed: parameters changed, or the
     /// view drifted / zoomed far enough that the current reference no longer
-    /// serves it well.
+    /// serves it well. Lambda in Set mode has a static fractal (doesn't depend
+    /// on center), so we skip center drift checks but allow zoom precision updates.
     fn should_request(&self) -> bool {
         let Some(key) = &self.last_request else {
             return true;
@@ -617,12 +641,19 @@ impl FractalApp {
         if key.julia != matches!(self.mode, FractalMode::Julia)
             || key.julia_c != self.julia_c
             || key.phoenix_p != self.phoenix_p
+            || key.lambda_l != self.lambda_l
             || key.iter != self.max_iterations
             || key.kind != self.kind
             || key.power != self.power
         {
             return true;
         }
+        // Lambda in Set mode is a static fractal; don't trigger recompute on center drift.
+        if self.kind == FractalKind::Lambda && matches!(self.mode, FractalMode::Mandelbrot) {
+            // But still recompute on significant zoom changes for precision
+            let ratio = self.view.half_height / key.half_height;
+            return !(0.5..=2.0).contains(&ratio);
+        }
         let ratio = self.view.half_height / key.half_height;
         self.drift_from(key) > 0.5 * self.view.half_height || !(0.5..=2.0).contains(&ratio)
     }
@@ -650,10 +681,16 @@ impl FractalApp {
     /// thread and pick up completed results. Web: compute inline.
     fn ensure_reference(&mut self) {
         if self.should_request() {
-            let key = self.current_key();
+            let mut key = self.current_key();
             let precision = self.view.precision_bits();
             let max_iter = key.iter.min(MAX_REF_POINTS as u32 - 1);
 
+            // Lambda in Set mode has a static fractal centered at origin.
+            if key.kind == FractalKind::Lambda && !key.julia {
+                key.center_re = big_from_f64(0.0, precision);
+                key.center_im = big_from_f64(0.0, precision);
+            }
+
             #[cfg(not(target_arch = "wasm32"))]
             {
                 self.worker.request(crate::worker::RefRequest {
@@ -667,6 +704,7 @@ impl FractalApp {
                     kind: key.kind,
                     power: key.power,
                     phoenix_p: key.phoenix_p,
+                    lambda_l: key.lambda_l,
                 });
                 self.pending = true;
             }
@@ -685,6 +723,7 @@ impl FractalApp {
                         key.kind,
                         key.power,
                         key.phoenix_p,
+                        key.lambda_l,
                     )
                 } else {
                     compute_set_reference(
@@ -695,6 +734,7 @@ impl FractalApp {
                         key.kind,
                         key.power,
                         key.phoenix_p,
+                        key.lambda_l,
                     )
                 };
                 self.apply_reference(
@@ -727,12 +767,13 @@ impl FractalApp {
             is_julia: matches!(self.mode, FractalMode::Julia) as u32,
             palette_id: self.palette,
             aa_level: if self.antialias { 2 } else { 1 },
-            kind: self.kind.shader_id(),
+            kind: self.kind as u32,
             power: self.power,
             dc_offset: self.dc_offset(),
             phoenix_p: [self.phoenix_p.0 as f32, self.phoenix_p.1 as f32],
+            lambda_l: [self.lambda_l.0 as f32, self.lambda_l.1 as f32],
             de_coloring: self.de_coloring as u32,
-            _pad: [0, 0, 0],
+            _pad: [0],
         }
     }
 
@@ -1046,10 +1087,11 @@ impl FractalApp {
     /// repaint while active. Animations render at full resolution/AA (they do not
     /// trigger the interaction low-res pass).
     fn tick_animations(&mut self, ui: &egui::Ui) {
-        // Julia c only matters in Julia mode; Phoenix p only for the Phoenix kind.
+        // Julia c only matters in Julia mode; Phoenix p only for the Phoenix kind; Lambda λ only for Lambda kind.
         let julia_on = self.anim.julia && self.mode == FractalMode::Julia;
         let phoenix_on = self.anim.phoenix && self.kind == FractalKind::Phoenix;
-        if !(self.anim.color || self.anim.zoom || julia_on || phoenix_on) {
+        let lambda_on = self.anim.lambda && self.kind == FractalKind::Lambda;
+        if !(self.anim.color || self.anim.zoom || julia_on || phoenix_on || lambda_on) {
             return;
         }
 
@@ -1076,6 +1118,15 @@ impl FractalApp {
                 self.anim.phoenix_base.1 + self.anim.phoenix_radius * s,
             );
         }
+        let lambda_on = self.anim.lambda && self.kind == FractalKind::Lambda;
+        if lambda_on {
+            self.anim.lambda_angle += std::f64::consts::TAU * self.anim.lambda_speed as f64 * dt;
+            let (s, c) = self.anim.lambda_angle.sin_cos();
+            self.lambda_l = (
+                self.anim.lambda_base.0 + self.anim.lambda_radius * c,
+                self.anim.lambda_base.1 + self.anim.lambda_radius * s,
+            );
+        }
         if self.anim.zoom && self.anim.zoom_speed != 0.0 {
             let min_hh = DEFAULT_HALF_HEIGHT * 1.0e-26; // practical f32-perturbation depth
             let max_hh = DEFAULT_HALF_HEIGHT * 4.0;
@@ -1124,6 +1175,22 @@ impl FractalApp {
                 ui.label("i");
             });
         }
+        if self.kind == FractalKind::Lambda {
+            ui.horizontal(|ui| {
+                ui.label("λ =");
+                ui.add(
+                    egui::DragValue::new(&mut self.lambda_l.0)
+                        .speed(0.001)
+                        .range(-2.0..=2.0),
+                );
+                ui.add(
+                    egui::DragValue::new(&mut self.lambda_l.1)
+                        .speed(0.001)
+                        .range(-2.0..=2.0),
+                );
+                ui.label("i");
+            });
+        }
         if self.kind != prev_kind {
             self.view = Self::default_view_for(self.mode, self.kind);
         }
@@ -1133,7 +1200,7 @@ impl FractalApp {
             ui.radio_value(&mut self.mode, FractalMode::Julia, "Julia");
         });
 
-        if self.mode == FractalMode::Julia {
+        if self.mode == FractalMode::Julia && self.kind != FractalKind::Lambda {
             ui.horizontal(|ui| {
                 ui.label("c =");
                 ui.add(
@@ -1271,6 +1338,24 @@ impl FractalApp {
                     );
                 }
             }
+            if self.kind == FractalKind::Lambda {
+                if ui.checkbox(&mut self.anim.lambda, "Morph λ").changed() && self.anim.lambda {
+                    self.anim.lambda_base = self.lambda_l;
+                    self.anim.lambda_angle = 0.0;
+                }
+                if self.anim.lambda {
+                    ui.add(
+                        egui::Slider::new(&mut self.anim.lambda_speed, 0.005..=0.5)
+                            .text("λ rev/s")
+                            .logarithmic(true),
+                    );
+                    ui.add(
+                        egui::Slider::new(&mut self.anim.lambda_radius, 0.005..=0.5)
+                            .text("λ radius")
+                            .logarithmic(true),
+                    );
+                }
+            }
         });
 
         ui.separator();

src/fractal/reference.rs

@@ -18,37 +18,23 @@ use crate::view::{Big, big_from_f64};
 #[derive(Clone, Copy, PartialEq, Eq, Debug)]
 pub enum FractalKind {
     /// `z -> z^2 + c`.
-    Mandelbrot,
+    Mandelbrot = 0,
     /// `z -> (|Re z| + i|Im z|)^2 + c`.
-    BurningShip,
+    BurningShip = 1,
     /// `z -> conj(z)^2 + c` (the Mandelbar).
-    Tricorn,
+    Tricorn = 2,
     /// `z -> z^power + c` (power >= 2).
-    Multibrot,
+    Multibrot = 3,
     /// `z -> |Re(z^2)| + i·Im(z^2) + c` (abs on the real output of the square).
-    Celtic,
+    Celtic = 4,
     /// `z -> (x^2 - y^2) - 2·x·|y|·i + c` (abs on the imaginary input).
-    Perpendicular,
+    Perpendicular = 5,
     /// `z -> |Re(z^2)| - |Im(z^2)|·i + c` (abs on both outputs).
-    Buffalo,
+    Buffalo = 6,
     /// `z -> z^2 + c + p·z_{n-1}` (two-term recurrence; `p` is `phoenix_p`).
-    Phoenix,
+    Phoenix = 7,
-}
+    /// `z -> lambda·z(1 - z)` (logistic map).
-
+    Lambda = 8,
-impl FractalKind {
-    /// Integer id matching the shader's `KIND_*` constants.
-    pub fn shader_id(self) -> u32 {
-        match self {
-            FractalKind::Mandelbrot => 0,
-            FractalKind::BurningShip => 1,
-            FractalKind::Tricorn => 2,
-            FractalKind::Multibrot => 3,
-            FractalKind::Celtic => 4,
-            FractalKind::Perpendicular => 5,
-            FractalKind::Buffalo => 6,
-            FractalKind::Phoenix => 7,
-        }
-    }
 }
 
 /// Reference orbit escapes once |Z|^2 exceeds this. Kept larger than the pixel
@@ -70,6 +56,7 @@ pub fn compute_reference(
     kind: FractalKind,
     power: u32,
     phoenix_p: (f64, f64),
+    lambda_l: (f64, f64),
 ) -> Vec<[f32; 2]> {
     let cr = c_re.clone().with_precision(precision).value();
     let ci = c_im.clone().with_precision(precision).value();
@@ -82,6 +69,9 @@ pub fn compute_reference(
     // Phoenix distortion constant `p` (a small fixed complex number).
     let pr = big_from_f64(phoenix_p.0, precision);
     let pi = big_from_f64(phoenix_p.1, precision);
+    // Lambda distortion constant `l` (a small fixed complex number).
+    let lr = big_from_f64(lambda_l.0, precision);
+    let li = big_from_f64(lambda_l.1, precision);
 
     let mut points: Vec<[f32; 2]> = Vec::with_capacity(max_iter as usize + 1);
 
@@ -144,6 +134,14 @@ pub fn compute_reference(
                 let pzi = &pr * &zi_prev + &pi * &zr_prev;
                 (re2 + &cr + pzr, im2 + &ci + pzi)
             }
+            FractalKind::Lambda => {
+                // λ·z(1 - z): logistic map.
+                let re2 = 1 - &zr;
+                let im2 = -&zi;
+                let lzr = &lr * &zr - &li * &zi;
+                let lzi = &lr * &zi + &li * &zr;
+                (&lzr * &re2 - &lzi * &im2, re2 * lzi + lzr * im2)
+            }
         };
 
         // Shift the previous iterate (only the Phoenix arm reads it).
@@ -182,6 +180,7 @@ fn complex_pow(zr: &Big, zi: &Big, power: u32, precision: usize) -> (Big, Big) {
 
 /// Convenience: parameter-plane ("Mandelbrot-set") reference (`z0 = 0`,
 /// `c = center`) for any `kind`.
+#[allow(clippy::too_many_arguments)]
 pub fn compute_set_reference(
     center_re: &Big,
     center_im: &Big,
@@ -190,10 +189,11 @@ pub fn compute_set_reference(
     kind: FractalKind,
     power: u32,
     phoenix_p: (f64, f64),
+    lambda_l: (f64, f64),
 ) -> Vec<[f32; 2]> {
     let zero = big_zero(precision);
     compute_reference(
-        &zero, &zero, center_re, center_im, max_iter, precision, kind, power, phoenix_p,
+        &zero, &zero, center_re, center_im, max_iter, precision, kind, power, phoenix_p, lambda_l,
     )
 }
 
@@ -207,8 +207,16 @@ mod tests {
     fn reference_matches_naive_f64() {
         let cr = Big::try_from(-0.75_f64).unwrap();
         let ci = Big::try_from(0.1_f64).unwrap();
-        let points =
+        let points = compute_set_reference(
-            compute_set_reference(&cr, &ci, 60, 200, FractalKind::Mandelbrot, 2, (0.0, 0.0));
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::Mandelbrot,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
 
         // Independent naive f64 orbit.
         let (c_re, c_im) = (-0.75_f64, 0.1_f64);
@@ -238,8 +246,16 @@ mod tests {
     fn interior_orbit_runs_full_length() {
         let cr = Big::try_from(-0.2_f64).unwrap();
         let ci = Big::try_from(0.0_f64).unwrap();
-        let points =
+        let points = compute_set_reference(
-            compute_set_reference(&cr, &ci, 500, 120, FractalKind::Mandelbrot, 2, (0.0, 0.0));
+            &cr,
+            &ci,
+            500,
+            120,
+            FractalKind::Mandelbrot,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
         assert_eq!(points.len(), 501, "interior orbit should not escape");
     }
 
@@ -248,8 +264,16 @@ mod tests {
     fn burning_ship_reference_matches_naive_f64() {
         let cr = Big::try_from(-1.75_f64).unwrap();
         let ci = Big::try_from(-0.03_f64).unwrap();
-        let points =
+        let points = compute_set_reference(
-            compute_set_reference(&cr, &ci, 60, 200, FractalKind::BurningShip, 2, (0.0, 0.0));
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::BurningShip,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
 
         let (c_re, c_im) = (-1.75_f64, -0.03_f64);
         let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
@@ -269,8 +293,16 @@ mod tests {
     fn multibrot3_reference_matches_naive_f64() {
         let cr = Big::try_from(0.3_f64).unwrap();
         let ci = Big::try_from(0.2_f64).unwrap();
-        let points =
+        let points = compute_set_reference(
-            compute_set_reference(&cr, &ci, 60, 200, FractalKind::Multibrot, 3, (0.0, 0.0));
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::Multibrot,
+            3,
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
 
         let (c_re, c_im) = (0.3_f64, 0.2_f64);
         let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
@@ -304,6 +336,7 @@ mod tests {
             FractalKind::Mandelbrot,
             2,
             (0.0, 0.0),
+            (0.0, 0.0),
         );
 
         let (mut zr, mut zi) = (0.15_f64, -0.1_f64);
@@ -324,7 +357,16 @@ mod tests {
     fn celtic_reference_matches_naive_f64() {
         let cr = Big::try_from(-0.6_f64).unwrap();
         let ci = Big::try_from(0.4_f64).unwrap();
-        let points = compute_set_reference(&cr, &ci, 60, 200, FractalKind::Celtic, 2, (0.0, 0.0));
+        let points = compute_set_reference(
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::Celtic,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
 
         let (c_re, c_im) = (-0.6_f64, 0.4_f64);
         let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
@@ -345,8 +387,16 @@ mod tests {
     fn perpendicular_reference_matches_naive_f64() {
         let cr = Big::try_from(-0.7_f64).unwrap();
         let ci = Big::try_from(-0.2_f64).unwrap();
-        let points =
+        let points = compute_set_reference(
-            compute_set_reference(&cr, &ci, 60, 200, FractalKind::Perpendicular, 2, (0.0, 0.0));
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::Perpendicular,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
 
         let (c_re, c_im) = (-0.7_f64, -0.2_f64);
         let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
@@ -367,7 +417,16 @@ mod tests {
     fn buffalo_reference_matches_naive_f64() {
         let cr = Big::try_from(-1.2_f64).unwrap();
         let ci = Big::try_from(-0.35_f64).unwrap();
-        let points = compute_set_reference(&cr, &ci, 60, 200, FractalKind::Buffalo, 2, (0.0, 0.0));
+        let points = compute_set_reference(
+            &cr,
+            &ci,
+            60,
+            200,
+            FractalKind::Buffalo,
+            2,
+            (0.0, 0.0),
+            (0.0, 0.0),
+        );
 
         let (c_re, c_im) = (-1.2_f64, -0.35_f64);
         let (mut zr, mut zi) = (0.0_f64, 0.0_f64);
@@ -389,7 +448,8 @@ mod tests {
         let cr = Big::try_from(0.5667_f64).unwrap();
         let ci = Big::try_from(0.0_f64).unwrap();
         let p = (-0.5_f64, 0.0_f64);
-        let points = compute_set_reference(&cr, &ci, 60, 200, FractalKind::Phoenix, 2, p);
+        let points =
+            compute_set_reference(&cr, &ci, 60, 200, FractalKind::Phoenix, 2, p, (0.0, 0.0));
 
         let (c_re, c_im) = (0.5667_f64, 0.0_f64);
         let (mut zr, mut zi) = (0.0_f64, 0.0_f64);

src/fractal/renderer.rs

@@ -78,10 +78,13 @@ pub struct Uniforms {
     /// ignored by other kinds. Kept next to `dc_offset` so both `vec2`s land on
     /// 8-byte boundaries, matching the shader's layout.
     pub phoenix_p: [f32; 2],
+    /// Distortion constant `l` for the Lambda map (`l·z(1 - z)`);
+    /// ignored by other kinds.
+    pub lambda_l: [f32; 2],
     /// 0 = escape-time coloring, 1 = distance-estimation shading.
     pub de_coloring: u32,
     /// Padding to a 16-byte multiple (uniform buffer requirement).
-    pub _pad: [u32; 3],
+    pub _pad: [u32; 1],
 }
 
 /// Offscreen textures for the two-pass render, recreated whenever the widget's

src/fractal/share.rs

@@ -23,6 +23,8 @@ pub struct ShareState {
     pub julia_c: (f64, f64),
     /// Distortion constant for the Phoenix kind (ignored by others).
     pub phoenix_p: (f64, f64),
+    /// Distortion constant for the Lambda kind (ignored by others).
+    pub lambda_l: (f64, f64),
     pub color_scale: f32,
     pub color_offset: f32,
     /// Palette index (`palette_id` in the shader).
@@ -44,6 +46,7 @@ impl ShareState {
                 FractalKind::Perpendicular => "perp",
                 FractalKind::Buffalo => "buffalo",
                 FractalKind::Phoenix => "phoenix",
+                FractalKind::Lambda => "lambda",
             }
         ));
         s.push_str(&format!("&pw={}", self.power));
@@ -53,6 +56,7 @@ impl ShareState {
         ));
         s.push_str(&format!("&jr={}&ji={}", self.julia_c.0, self.julia_c.1));
         s.push_str(&format!("&px={}&py={}", self.phoenix_p.0, self.phoenix_p.1));
+        s.push_str(&format!("&lx={}&ly={}", self.lambda_l.0, self.lambda_l.1));
         s.push_str(&format!(
             "&cs={}&co={}&pal={}",
             self.color_scale, self.color_offset, self.palette
@@ -82,6 +86,7 @@ impl ShareState {
                     "perp" => FractalKind::Perpendicular,
                     "buffalo" => FractalKind::Buffalo,
                     "phoenix" => FractalKind::Phoenix,
+                    "lambda" => FractalKind::Lambda,
                     _ => FractalKind::Mandelbrot,
                 })
                 .unwrap_or(FractalKind::Mandelbrot),
@@ -98,6 +103,10 @@ impl ShareState {
                 map.get("px").and_then(|s| s.parse().ok()).unwrap_or(-0.5),
                 map.get("py").and_then(|s| s.parse().ok()).unwrap_or(0.0),
             ),
+            lambda_l: (
+                map.get("lx").and_then(|s| s.parse().ok()).unwrap_or(-0.5),
+                map.get("ly").and_then(|s| s.parse().ok()).unwrap_or(0.0),
+            ),
             color_scale: map.get("cs").and_then(|s| s.parse().ok()).unwrap_or(0.02),
             color_offset: map.get("co").and_then(|s| s.parse().ok()).unwrap_or(0.0),
             palette: map.get("pal").and_then(|s| s.parse().ok()).unwrap_or(0),
@@ -121,6 +130,7 @@ mod tests {
             iterations: 4000,
             julia_c: (-0.123, 0.745),
             phoenix_p: (-0.5, 0.1),
+            lambda_l: (-0.5, 0.0),
             color_scale: 0.02,
             color_offset: 0.25,
             palette: 3,

src/shaders/mandelbrot.wgsl

@@ -30,6 +30,9 @@ struct Uniforms {
     // Distortion constant p for the Phoenix map (z^2 + c + p*z_{n-1}); unused
     // by other kinds. Placed by dc_offset so both vec2s stay 8-byte aligned.
     phoenix_p: vec2<f32>,
+    // Distortion constant l for the Lambda map (l*z(1 - z_{n-1})); unused
+    // by other kinds.
+    lambda_l: vec2<f32>,
     // 0 = escape-time coloring, 1 = distance-estimation shading.
     de_coloring: u32,
 };
@@ -42,6 +45,7 @@ const KIND_CELTIC: u32 = 4u;
 const KIND_PERPENDICULAR: u32 = 5u;
 const KIND_BUFFALO: u32 = 6u;
 const KIND_PHOENIX: u32 = 7u;
+const KIND_LAMBDA: u32 = 8u;
 
 @group(0) @binding(0) var<uniform> u: Uniforms;
 @group(0) @binding(1) var<storage, read> ref_orbit: array<vec2<f32>>;
@@ -82,7 +86,7 @@ fn conj(a: vec2<f32>) -> vec2<f32> {
 // the sign flips that happen all along the axes, where the ship's detail lives.
 fn diffabs(c: f32, d: f32) -> f32 {
     let cd = c + d;
-    if (c >= 0.0) {
+    if c >= 0.0 {
         return select(-(2.0 * c + d), d, cd >= 0.0);
     }
     return select(-d, 2.0 * c + d, cd > 0.0);
@@ -120,7 +124,7 @@ fn multibrot_delta(z: vec2<f32>, e: vec2<f32>, p: u32) -> vec2<f32> {
 // where `z` is the reference orbit value X_m. `step_add` (dc) is added by the
 // caller. Must match `FractalKind` on the CPU side.
 fn advance_delta(z: vec2<f32>, e: vec2<f32>) -> vec2<f32> {
-    if (u.kind == KIND_BURNING_SHIP) {
+    if u.kind == KIND_BURNING_SHIP {
         // (|x| + i|y|)^2 has real part x^2 - y^2 (an ordinary square delta) and
         // imaginary part 2|x y|. The imaginary delta is 2(|x y| - |X Y|); diffabs
         // computes it exactly, even where the product x y changes sign — which the
@@ -129,29 +133,33 @@ fn advance_delta(z: vec2<f32>, e: vec2<f32>) -> vec2<f32> {
         let base = 2.0 * cmul(z, e) + cmul(e, e);
         let dp = z.x * e.y + z.y * e.x + e.x * e.y;
         return vec2<f32>(base.x, 2.0 * diffabs(z.x * z.y, dp));
-    } else if (u.kind == KIND_TRICORN) {
+    } else if u.kind == KIND_TRICORN {
         let cz = conj(z);
         let ce = conj(e);
         return 2.0 * cmul(cz, ce) + cmul(ce, ce);
-    } else if (u.kind == KIND_MULTIBROT) {
+    } else if u.kind == KIND_MULTIBROT {
         return multibrot_delta(z, e, clamp(u.power, 2u, 8u));
-    } else if (u.kind == KIND_CELTIC) {
+    } else if u.kind == KIND_CELTIC {
         // z^2 delta split: sq.x = delta of Re(z^2), sq.y = delta of Im(z^2).
         // Celtic abs the real output, so |Re(z^2)| delta = diffabs(Re(Z^2), sq.x).
         let sq = 2.0 * cmul(z, e) + cmul(e, e);
         return vec2<f32>(diffabs(z.x * z.x - z.y * z.y, sq.x), sq.y);
-    } else if (u.kind == KIND_BUFFALO) {
+    } else if u.kind == KIND_BUFFALO {
         // Abs both outputs: real |Re(z^2)|, imag -|Im(z^2)| (Im(Z^2) = 2 X Y).
         let sq = 2.0 * cmul(z, e) + cmul(e, e);
         return vec2<f32>(diffabs(z.x * z.x - z.y * z.y, sq.x),
                          -diffabs(2.0 * z.x * z.y, sq.y));
-    } else if (u.kind == KIND_PERPENDICULAR) {
+    } else if u.kind == KIND_PERPENDICULAR {
         // real x^2 - y^2 (ordinary square delta), imag -2 x |y|.
         // d(-2 x |y|) = -2[ X·(|Y+ey|-|Y|) + ex·|Y+ey| ]; diffabs gives |Y+ey|-|Y|.
         let sq = 2.0 * cmul(z, e) + cmul(e, e);
         let da = diffabs(z.y, e.y);        // |Y + ey| - |Y|
         let abs_yf = abs(z.y) + da;        // |Y + ey|
         return vec2<f32>(sq.x, -2.0 * (z.x * da + e.x * abs_yf));
+    } else if u.kind == KIND_LAMBDA {
+        // Lambda map: z^{n+1} = λ·z·(1-z). Delta: e = λ·e·(1-2z-e).
+        let one_minus_2z_minus_e = vec2<f32>(1.0 - 2.0*z.x - e.x, -2.0*z.y - e.y);
+        return cmul(u.lambda_l, cmul(e, one_minus_2z_minus_e));
     }
     return 2.0 * cmul(z, e) + cmul(e, e); // Mandelbrot (and Phoenix square part)
 }
@@ -162,31 +170,34 @@ fn advance_delta(z: vec2<f32>, e: vec2<f32>) -> vec2<f32> {
 // Burning Ship / Tricorn we use |f'| ~ |2Z|, which keeps the DE magnitude close
 // enough to de-speckle filaments.
 fn fprime(z: vec2<f32>) -> vec2<f32> {
-    if (u.kind == KIND_MULTIBROT) {
+    if u.kind == KIND_MULTIBROT {
         let p = clamp(u.power, 2u, 8u);
         var zk = vec2<f32>(1.0, 0.0); // Z^0
         for (var k: u32 = 1u; k < p; k = k + 1u) {
             zk = cmul(zk, z); // -> Z^{p-1}
         }
         return f32(p) * zk;
+    } else if u.kind == KIND_LAMBDA {
+        // Lambda: f'(z) = λ·(1-2z).
+        return cmul(u.lambda_l, vec2<f32>(1.0 - 2.0*z.x, -2.0*z.y));
     }
     return 2.0 * z;
 }
 
 // Smooth cyclic palettes (Inigo Quilez cosine palettes), selected by id.
 fn palette(id: u32, t: f32) -> vec3<f32> {
-    if (id == 4u) {
+    if id == 4u {
         return vec3<f32>(t, t, t); // grayscale
     }
     let a = vec3<f32>(0.5, 0.5, 0.5);
     let b = vec3<f32>(0.5, 0.5, 0.5);
     var c = vec3<f32>(1.0, 1.0, 1.0);
     var d = vec3<f32>(0.00, 0.10, 0.20); // 0: amber / blue
-    if (id == 1u) {
+    if id == 1u {
         d = vec3<f32>(0.00, 0.33, 0.67); // rainbow
-    } else if (id == 2u) {
+    } else if id == 2u {
         d = vec3<f32>(0.30, 0.20, 0.20); // warm ember
-    } else if (id == 3u) {
+    } else if id == 3u {
         c = vec3<f32>(1.0, 1.0, 0.5);
         d = vec3<f32>(0.80, 0.90, 0.30); // lime / magenta
     }
@@ -220,7 +231,7 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
     // y_{n-1}, and its derivative for DE). Both start at 0 (y_{-1} = 0).
     var e_prev = vec2<f32>(0.0, 0.0);
     var dz_prev = vec2<f32>(0.0, 0.0);
-    if (u.is_julia != 0u) {
+    if u.is_julia != 0u {
         step_add = vec2<f32>(0.0, 0.0);
         e = offset;
         dz = vec2<f32>(1.0, 0.0);
@@ -237,20 +248,20 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
             z = xm + e;
 
             let z2 = dot(z, z);
-        if (z2 > u.bailout_sq) {
+            if z2 > u.bailout_sq {
                 escaped = true;
                 break;
             }
-        if (n >= u.max_iter) {
+            if n >= u.max_iter {
                 break; // interior
             }
 
         // Propagate the derivative of the full orbit (unaffected by rebasing,
         // which only re-expresses the same value). Only when DE is enabled.
         // Phoenix's two-term map adds p·dz_{n-1} and carries the previous dz.
-        if (u.de_coloring != 0u) {
+            if u.de_coloring != 0u {
                 var dz_new = cmul(fprime(z), dz) + dz_seed;
-            if (u.kind == KIND_PHOENIX) {
+                if u.kind == KIND_PHOENIX {
                     dz_new = dz_new + cmul(u.phoenix_p, dz_prev);
                     dz_prev = dz;
                 }
@@ -261,7 +272,7 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
         // Phoenix additionally adds p·e_{n-1} and carries the previous delta.
             let e_old = e;
             e = advance_delta(xm, e) + step_add;
-        if (u.kind == KIND_PHOENIX) {
+            if u.kind == KIND_PHOENIX {
                 e = e + cmul(u.phoenix_p, e_prev);
                 e_prev = e_old;
             }
@@ -269,7 +280,7 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
             n = n + 1u;
 
         // Keep the reference index valid and the delta small.
-        if (m >= u.ref_len) {
+            if m >= u.ref_len {
             // Reference exhausted: any pixel that followed it this far has
             // effectively escaped (interior pixels rebase before reaching here).
                 z = ref_orbit[u.ref_len - 1u] + e;
@@ -277,12 +288,12 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
                 break;
             }
             let y = ref_orbit[m] + e;
-        if (dot(y, y) < dot(e, e)) {
+            if dot(y, y) < dot(e, e) {
             // Rebase to index 0: carry the full value as the new delta. Valid
             // because y_n = X[0] + (y_n - X[0]); for Mandelbrot X[0]=0.
             // Phoenix: after rebasing the implied previous reference is Y[-1]=0,
             // so the previous delta becomes the full previous value y_n (= z).
-            if (u.kind == KIND_PHOENIX) {
+                if u.kind == KIND_PHOENIX {
                     e_prev = z;
                 }
                 e = y - z0;
@@ -290,7 +301,7 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
             }
         }
 
-    if (!escaped) {
+    if !escaped {
         return Sample(0.0, 1.0, false); // interior of the set
     }
 
@@ -306,7 +317,7 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
     let ci = sqrt(max(smooth_i, 0.0));
 
     var de = 1.0;
-    if (u.de_coloring != 0u) {
+    if u.de_coloring != 0u {
         // Exterior distance estimate (complex-plane units): |z|·ln|z| / |dz|.
         // Divided by the pixel footprint it becomes a distance in pixels; we
         // darken toward the boundary (< ~1 px away) so filaments stay crisp
@@ -324,7 +335,7 @@ fn iterate_sample(offset: vec2<f32>, px: f32) -> Sample {
 // only color-dependent step, so it can be redone without re-iterating. Interior
 // samples are black.
 fn color_sample(s: Sample) -> vec3<f32> {
-    if (!s.escaped) {
+    if !s.escaped {
         return vec3<f32>(0.0, 0.0, 0.0);
     }
     let t = fract(s.ci * u.color_scale + u.color_offset);
@@ -353,7 +364,7 @@ fn fs_data(in: VsOut) -> @location(0) vec4<f32> {
             let jx = (f32(sx) + 0.5) * inv - 0.5;
             let jy = (f32(sy) + 0.5) * inv - 0.5;
             let s = iterate_sample(base + jx * dx + jy * dy, px);
-            if (s.escaped) {
+            if s.escaped {
                 ci_sum = ci_sum + s.ci;
                 de_sum = de_sum + s.de;
                 escaped_n = escaped_n + 1u;

src/worker.rs

@@ -24,6 +24,8 @@ pub struct RefRequest {
     pub power: u32,
     /// Distortion constant for the Phoenix map (ignored by other kinds).
     pub phoenix_p: (f64, f64),
+    /// Distortion constant for the Lambda map (ignored by other kinds).
+    pub lambda_l: (f64, f64),
 }
 
 pub struct RefResult {
@@ -104,6 +106,7 @@ fn compute(req: &RefRequest) -> Vec<[f32; 2]> {
             req.kind,
             req.power,
             req.phoenix_p,
+            req.lambda_l,
         )
     } else {
         compute_set_reference(
@@ -114,6 +117,7 @@ fn compute(req: &RefRequest) -> Vec<[f32; 2]> {
             req.kind,
             req.power,
             req.phoenix_p,
+            req.lambda_l,
         )
     }
 }