diff --git a/egui_demo_lib/src/apps/demo/dancing_strings.rs b/egui_demo_lib/src/apps/demo/dancing_strings.rs
index 2d7245cc..e277d196 100644
--- a/egui_demo_lib/src/apps/demo/dancing_strings.rs
+++ b/egui_demo_lib/src/apps/demo/dancing_strings.rs
@@ -34,6 +34,9 @@ impl super::View for DancingStrings {
             let desired_size = ui.available_width() * vec2(1.0, 0.35);
             let (_id, rect) = ui.allocate_space(desired_size);
 
+            let to_screen =
+                math::RectTransform::from_to(Rect::from_x_y_ranges(0.0..=1.0, -1.0..=1.0), rect);
+
             let mut shapes = vec![];
 
             for &mode in &[2, 3, 5] {
@@ -46,11 +49,7 @@ impl super::View for DancingStrings {
                         let t = i as f32 / (n as f32);
                         let amp = (time as f32 * speed * mode).sin() / mode;
                         let y = amp * (t * std::f32::consts::TAU / 2.0 * mode).sin();
-
-                        pos2(
-                            lerp(rect.x_range(), t),
-                            remap(y, -1.0..=1.0, rect.y_range()),
-                        )
+                        to_screen * pos2(t, y)
                     })
                     .collect();
 
diff --git a/egui_demo_lib/src/apps/demo/painting.rs b/egui_demo_lib/src/apps/demo/painting.rs
index 24b90d33..483bd9cc 100644
--- a/egui_demo_lib/src/apps/demo/painting.rs
+++ b/egui_demo_lib/src/apps/demo/painting.rs
@@ -3,7 +3,8 @@ use egui::*;
 #[cfg_attr(feature = "persistence", derive(serde::Deserialize, serde::Serialize))]
 #[cfg_attr(feature = "persistence", serde(default))]
 pub struct Painting {
-    lines: Vec<Vec<Vec2>>,
+    /// in 0-1 normalized coordinates
+    lines: Vec<Vec<Pos2>>,
     stroke: Stroke,
 }
 
@@ -32,6 +33,12 @@ impl Painting {
             ui.allocate_painter(ui.available_size_before_wrap_finite(), Sense::drag());
         let rect = response.rect;
 
+        let to_screen = emath::RectTransform::from_to(
+            Rect::from_min_size(Pos2::ZERO, rect.square_proportions()),
+            rect,
+        );
+        let from_screen = to_screen.inverse();
+
         if self.lines.is_empty() {
             self.lines.push(vec![]);
         }
@@ -39,7 +46,7 @@ impl Painting {
         let current_line = self.lines.last_mut().unwrap();
 
         if let Some(pointer_pos) = response.interact_pointer_pos() {
-            let canvas_pos = pointer_pos - rect.min;
+            let canvas_pos = from_screen * pointer_pos;
             if current_line.last() != Some(&canvas_pos) {
                 current_line.push(canvas_pos);
             }
@@ -49,7 +56,7 @@ impl Painting {
 
         for line in &self.lines {
             if line.len() >= 2 {
-                let points: Vec<Pos2> = line.iter().map(|p| rect.min + *p).collect();
+                let points: Vec<Pos2> = line.iter().map(|p| to_screen * *p).collect();
                 painter.add(Shape::line(points, self.stroke));
             }
         }
diff --git a/egui_demo_lib/src/apps/fractal_clock.rs b/egui_demo_lib/src/apps/fractal_clock.rs
index 54c7f554..a174e7e2 100644
--- a/egui_demo_lib/src/apps/fractal_clock.rs
+++ b/egui_demo_lib/src/apps/fractal_clock.rs
@@ -100,8 +100,6 @@ impl FractalClock {
     }
 
     fn paint(&mut self, painter: &Painter) {
-        let rect = painter.clip_rect();
-
         struct Hand {
             length: f32,
             angle: f32,
@@ -130,14 +128,18 @@ impl FractalClock {
             Hand::from_length_angle(0.5, angle_from_period(12.0 * 60.0 * 60.0)),
         ];
 
-        let scale = self.zoom * rect.width().min(rect.height());
         let mut shapes: Vec<Shape> = Vec::new();
-        let mut paint_line = |points: [Pos2; 2], color: Color32, width: f32| {
-            let line = [
-                rect.center() + scale * points[0].to_vec2(),
-                rect.center() + scale * points[1].to_vec2(),
-            ];
 
+        let rect = painter.clip_rect();
+        let to_screen = emath::RectTransform::from_to(
+            Rect::from_center_size(Pos2::ZERO, rect.square_proportions() / self.zoom),
+            rect,
+        );
+
+        let mut paint_line = |points: [Pos2; 2], color: Color32, width: f32| {
+            let line = [to_screen * points[0], to_screen * points[1]];
+
+            // culling
             if rect.intersects(Rect::from_two_pos(line[0], line[1])) {
                 shapes.push(Shape::line_segment(line, (width, color)));
             }
@@ -186,6 +188,9 @@ impl FractalClock {
             width *= self.width_factor;
 
             let luminance_u8 = (255.0 * luminance).round() as u8;
+            if luminance_u8 == 0 {
+                break;
+            }
 
             for &rotor in &hand_rotors {
                 for a in &nodes {
diff --git a/egui_demo_lib/src/frame_history.rs b/egui_demo_lib/src/frame_history.rs
index 0939e0cf..5d8748ef 100644
--- a/egui_demo_lib/src/frame_history.rs
+++ b/egui_demo_lib/src/frame_history.rs
@@ -58,7 +58,6 @@ impl FrameHistory {
     fn graph(&mut self, ui: &mut egui::Ui) -> egui::Response {
         use egui::*;
 
-        let graph_top_cpu_usage = 0.010;
         ui.label("egui CPU usage history");
 
         let history = &self.frame_times;
@@ -69,12 +68,17 @@ impl FrameHistory {
         let (rect, response) = ui.allocate_at_least(size, Sense::hover());
         let style = ui.style().noninteractive();
 
-        let mut shapes = vec![Shape::Rect {
+        let graph_top_cpu_usage = 0.010;
+        let graph_rect = Rect::from_x_y_ranges(history.max_age()..=0.0, graph_top_cpu_usage..=0.0);
+        let to_screen = emath::RectTransform::from_to(graph_rect, rect);
+
+        let mut shapes = Vec::with_capacity(3 + 2 * history.len());
+        shapes.push(Shape::Rect {
             rect,
             corner_radius: style.corner_radius,
             fill: ui.visuals().extreme_bg_color,
             stroke: ui.style().noninteractive().bg_stroke,
-        }];
+        });
 
         let rect = rect.shrink(4.0);
         let color = ui.visuals().text_color();
@@ -87,7 +91,7 @@ impl FrameHistory {
                     [pos2(rect.left(), y), pos2(rect.right(), y)],
                     line_stroke,
                 ));
-                let cpu_usage = remap(y, rect.bottom_up_range(), 0.0..=graph_top_cpu_usage);
+                let cpu_usage = to_screen.inverse().transform_pos(pointer_pos).y;
                 let text = format!("{:.1} ms", 1e3 * cpu_usage);
                 shapes.push(Shape::text(
                     ui.fonts(),
@@ -106,16 +110,15 @@ impl FrameHistory {
 
         for (time, cpu_usage) in history.iter() {
             let age = (right_side_time - time) as f32;
-            let x = remap(age, history.max_age()..=0.0, rect.x_range());
-            let y = remap_clamp(cpu_usage, 0.0..=graph_top_cpu_usage, rect.bottom_up_range());
+            let pos = to_screen.transform_pos_clamped(Pos2::new(age, cpu_usage));
 
             shapes.push(Shape::line_segment(
-                [pos2(x, rect.bottom()), pos2(x, y)],
+                [pos2(pos.x, rect.bottom()), pos],
                 line_stroke,
             ));
 
             if cpu_usage < graph_top_cpu_usage {
-                shapes.push(Shape::circle_filled(pos2(x, y), radius, circle_color));
+                shapes.push(Shape::circle_filled(pos, radius, circle_color));
             }
         }
 
diff --git a/emath/src/lib.rs b/emath/src/lib.rs
index 37742d91..23f121b0 100644
--- a/emath/src/lib.rs
+++ b/emath/src/lib.rs
@@ -60,6 +60,7 @@ use std::ops::{Add, Div, Mul, RangeInclusive, Sub};
 pub mod align;
 mod pos2;
 mod rect;
+mod rect_transform;
 mod rot2;
 pub mod smart_aim;
 mod vec2;
@@ -68,6 +69,7 @@ pub use {
     align::{Align, Align2},
     pos2::*,
     rect::*,
+    rect_transform::*,
     rot2::*,
     vec2::*,
 };
diff --git a/emath/src/rect.rs b/emath/src/rect.rs
index e98555ff..fe4fc6f4 100644
--- a/emath/src/rect.rs
+++ b/emath/src/rect.rs
@@ -202,6 +202,29 @@ impl Rect {
     pub fn height(&self) -> f32 {
         self.max.y - self.min.y
     }
+
+    /// Width / height
+    ///
+    /// * `aspect_ratio < 1`: portrait / high
+    /// * `aspect_ratio = 1`: square
+    /// * `aspect_ratio > 1`: landscape / wide
+    pub fn aspect_ratio(&self) -> f32 {
+        self.width() / self.height()
+    }
+
+    /// `[2, 1]` for wide screen, and `[1, 2]` for portrait, etc.
+    /// At least one dimension = 1, the other >= 1
+    /// Returns the proportions required to letter-box a square view area.
+    pub fn square_proportions(&self) -> Vec2 {
+        let w = self.width();
+        let h = self.height();
+        if w > h {
+            vec2(w / h, 1.0)
+        } else {
+            vec2(1.0, h / w)
+        }
+    }
+
     pub fn area(&self) -> f32 {
         self.width() * self.height()
     }
diff --git a/emath/src/rect_transform.rs b/emath/src/rect_transform.rs
new file mode 100644
index 00000000..e758652c
--- /dev/null
+++ b/emath/src/rect_transform.rs
@@ -0,0 +1,73 @@
+use crate::*;
+
+/// Linearly transforms positions from one [`Rect`] to another.
+///
+/// `RectTransform` stores the rectangles, and therefore supports clamping and culling.
+#[derive(Clone, Copy, Debug, PartialEq)]
+pub struct RectTransform {
+    from: Rect,
+    to: Rect,
+}
+
+impl RectTransform {
+    pub fn identity(from_and_to: Rect) -> Self {
+        Self::from_to(from_and_to, from_and_to)
+    }
+
+    pub fn from_to(from: Rect, to: Rect) -> Self {
+        Self { from, to }
+    }
+
+    pub fn from(&self) -> &Rect {
+        &self.from
+    }
+
+    pub fn to(&self) -> &Rect {
+        &self.to
+    }
+
+    pub fn inverse(&self) -> RectTransform {
+        Self::from_to(self.to, self.from)
+    }
+
+    /// Transforms the given coordinate in the `from` space to the `to` space.
+    pub fn transform_pos(&self, pos: Pos2) -> Pos2 {
+        pos2(
+            remap(pos.x, self.from.x_range(), self.to.x_range()),
+            remap(pos.y, self.from.y_range(), self.to.y_range()),
+        )
+    }
+
+    /// Transforms the given rectangle in the `in`-space to a rectangle in the `out`-space.
+    pub fn transform_rect(&self, rect: Rect) -> Rect {
+        Rect {
+            min: self.transform_pos(rect.min),
+            max: self.transform_pos(rect.max),
+        }
+    }
+
+    /// Transforms the given coordinate in the `from` space to the `to` space,
+    /// clamping if necessary.
+    pub fn transform_pos_clamped(&self, pos: Pos2) -> Pos2 {
+        pos2(
+            remap_clamp(pos.x, self.from.x_range(), self.to.x_range()),
+            remap_clamp(pos.y, self.from.y_range(), self.to.y_range()),
+        )
+    }
+}
+
+/// Transforms the position.
+impl std::ops::Mul<Pos2> for RectTransform {
+    type Output = Pos2;
+    fn mul(self, pos: Pos2) -> Pos2 {
+        self.transform_pos(pos)
+    }
+}
+
+/// Transforms the position.
+impl std::ops::Mul<Pos2> for &RectTransform {
+    type Output = Pos2;
+    fn mul(self, pos: Pos2) -> Pos2 {
+        self.transform_pos(pos)
+    }
+}
diff --git a/emath/src/rot2.rs b/emath/src/rot2.rs
index 2ec1783d..e274bd85 100644
--- a/emath/src/rot2.rs
+++ b/emath/src/rot2.rs
@@ -8,7 +8,9 @@ use super::Vec2;
 // `vec2(c,s)` represents where the X axis will end up after rotation.
 //
 /// Represents a rotation in the 2D plane.
+//
 /// A rotation of 𝞃/4 = 90° rotates the X axis to the Y axis.
+//
 /// Normally a `Rot2` is normalized (unit-length).
 /// If not, it will also scale vectors.
 #[derive(Clone, Copy, PartialEq)]
@@ -104,6 +106,7 @@ impl std::ops::Mul<Rot2> for Rot2 {
     }
 }
 
+/// Rotates (and maybe scales) the vector.
 impl std::ops::Mul<Vec2> for Rot2 {
     type Output = Vec2;
     fn mul(self, v: Vec2) -> Vec2 {
@@ -114,6 +117,7 @@ impl std::ops::Mul<Vec2> for Rot2 {
     }
 }
 
+/// Scales the rotor.
 impl std::ops::Mul<Rot2> for f32 {
     type Output = Rot2;
     fn mul(self, r: Rot2) -> Rot2 {
@@ -124,6 +128,7 @@ impl std::ops::Mul<Rot2> for f32 {
     }
 }
 
+/// Scales the rotor.
 impl std::ops::Mul<f32> for Rot2 {
     type Output = Rot2;
     fn mul(self, r: f32) -> Rot2 {
@@ -134,6 +139,7 @@ impl std::ops::Mul<f32> for Rot2 {
     }
 }
 
+/// Scales the rotor.
 impl std::ops::Div<f32> for Rot2 {
     type Output = Rot2;
     fn div(self, r: f32) -> Rot2 {