bezier: Improve cubic-t-for-distance (#338)

Fixes a _major_ precision bug in `cubic-t-for-distance` and
`cubic-shorten` leading to bad mark/anchor placement on Bézier paths.

Also uses correct distance based placement for `point-on-path` if Bézier
curves are involved!

src/bezier.typ

@@ -291,27 +291,31 @@
 /// start s, if d is negative, it starts form the curves end
 /// e.
 /// -> float Bezier t value from [0,1]
-#let cubic-t-for-distance(s, e, c1, c2, d, samples: 10) = {
+#let cubic-t-for-distance(s, e, c1, c2, d, samples: 20) = {
+  let travel-forwards(s, e, c1, c2, d) = {
+    let sum = 0
+    for n in range(1, samples + 1) {
+      let t0 = (n - 1) / samples
+      let t1 = n / samples
+
+      let segment-dist = vector.dist(cubic-point(s, e, c1, c2, t0),
+                                     cubic-point(s, e, c1, c2, t1))
+      if sum <= d and d <= sum + segment-dist {
+        return t0 + (d - sum) / segment-dist / samples
+      }
+      sum += segment-dist
+    }
+    return 1
+  }
+
   if d == 0 {
     return 0
   }
 
   if d > 0 {
-    let travel = 0 // Distance traveled along the curve
-    let last = s
-    for t in range(1, samples + 1) {
-      let t = t / samples
-      let curr = cubic-point(s, e, c1, c2, t)
-      let dist = vector.dist(last, curr)
-      travel += dist
-      if travel >= d {
-        return t - 1/samples + d / (travel * samples)
-      }
-      last = curr
-    }
-    return 1
+    return travel-forwards(s, e, c1, c2, d)
   } else {
-    return 1 - cubic-t-for-distance(e, s, c2, c1, -d, samples: samples)
+    return 1 - travel-forwards(e, s, c2, c1, -d)
   }
 }
 
@@ -330,34 +334,14 @@
 /// - samples (int): Maximum of samples/steps to use
 /// -> (s, e, c1, c2) Shortened curve
 #let cubic-shorten(s, e, c1, c2, d, samples: 15) = {
-  if d == 0 {
-    return (s, e, c1, c2)
-  }
+  if d == 0 { return (s, e, c1, c2) }
 
-  let split-t = 0
-  if d > 0 {
-    let travel = 0 // Distance traveled along the curve
-
-    let last = s
-    for t in range(1, samples + 1) {
-      let t = t / samples
-      let curr = cubic-point(s, e, c1, c2, t)
-      let dist = vector.dist(last, curr)
-      travel += dist
-      if travel >= d {
-        split-t = t - 1/samples + d / (travel * samples)
-        break
-      }
-      last = curr
-    }
+  let (left, right) = split(s, e, c1, c2, cubic-t-for-distance(s, e, c1, c2, d, samples: samples))
+  return if d > 0 {
+    right
   } else {
-    // Run the algorithm from end to start by swapping the curve.
-    let (e, s, c2, c1) = cubic-shorten(e, s, c2, c1, -d, samples: samples)
-    return (s, e, c1, c2)
+    left
   }
-
-  let (_, right) = split(s, e, c1, c2, split-t)
-  return right
 }
 
 /// Align curve points pts to the line start-end

src/path-util.typ

@@ -54,29 +54,18 @@
 /// -> float: Length of the segment in canvas units
 #let segment-length(s) = {
   let samples = default-samples
-  let type = s.at(0)
-  let pts = ()
+  let (type, ..pts) = s
   if type == "line" {
-    pts = s.slice(1)
+    let len = 0
+    for i in range(1, pts.len()) {
+      len += vector.len(vector.sub(pts.at(i - 1), pts.at(i)))
+    }
+    return len
   } else if type == "cubic" {
-    let (a, b, c, d) = s.slice(1)
-    pts.push(a)
-    pts = range(1, samples).map(t =>
-      bezier.cubic-point(a, b, c, d, t / samples))
-    pts.push(b)
+    return bezier.cubic-arclen(..pts, samples: samples)
   } else {
-    panic("Not implemented")
-  }
-
-  let l = 0
-
-  let pt = pts.at(0)
-  for i in range(1, pts.len()) {
-    l += vector.len(vector.sub(pts.at(i), pt))
-    pt = pts.at(i)
+    panic("Invalid segment: " + type)
   }
-
-  return l
 }
 
 /// Find point at position on polyline segment
@@ -96,13 +85,9 @@
     return s.at(1)
   }
 
-  let dist = (a, b) => {
-    vector.len(vector.sub(b, a))
-  }
-
   let traveled-length = 0
   for i in range(2, s.len()) {
-    let part-length = dist(s.at(i - 1), s.at(i))
+    let part-length = vector.dist(s.at(i - 1), s.at(i))
 
     if traveled-length / l <= t and (traveled-length + part-length) / l >= t {
       let f = (t - traveled-length / l) / (part-length / l)
@@ -124,12 +109,12 @@
 /// - t (float): Position (from 0 to 1)
 /// -> vector: Position on segment
 #let point-on-segment(s, t) = {
-  let type = s.at(0)
+  let (type, ..pts) = s
   if type == "line" {
     return point-on-polyline(s, t)
   } else if type == "cubic" {
-    let (a, b, c, d) = s.slice(1)
-    return bezier.cubic-point(a, b, c, d, t)
+    let len = bezier.cubic-arclen(..pts) * calc.min(calc.max(0, t), 1)
+    return bezier.cubic-point(..pts, bezier.cubic-t-for-distance(..pts, len))
   }
 }