Add extremities finding for cubic bezier

examples/cubic_bezier.cr

@@ -8,6 +8,8 @@ module PF
     CTL_COLOR   = Pixel.new(0x505050FF)
     CURVE_COLOR = Pixel.new(0x0077FFFF)
     SEL_COLOR   = Pixel.new(0xFFFF00FF)
+    EXT_X_COLOR = Pixel.new(0xFF00FFFF)
+    EXT_Y_COLOR = Pixel.new(0x00FF00FF)
 
     @curve : Bezier::Cubic(Float64)
 
@@ -57,6 +59,16 @@ module PF
       draw_string("Length: " + @curve.length.round(2).to_s, 5, 5, FONT_COLOR)
       draw_curve(@curve, CURVE_COLOR)
 
+      ext_x, ext_y = @curve.extremeties
+
+      ext_x.each do |point|
+        draw_circle(point.to_i, 3, EXT_X_COLOR)
+      end
+
+      ext_y.each do |point|
+        draw_circle(point.to_i, 3, EXT_Y_COLOR)
+      end
+
       fill_circle(@curve.p0.to_i, 2, POINT_COLOR)
       fill_circle(@curve.p1.to_i, 2, POINT_COLOR)
       fill_circle(@curve.p2.to_i, 2, POINT_COLOR)

src/bezier/cubic.cr

@@ -17,6 +17,32 @@ module PF
         6 * (1 - t) * (p2 - 2 * p1 + p0) + 6 * t * (p3 - 2 * p2 + p1)
       end
 
+      def self.extremeties(p0 : Number, p1 : Number, p2 : Number, p3 : Number)
+        a = 3 * p3 - 9 * p2 + 9 * p1 - 3 * p0
+        b = 6 * p0 - 12 * p1 + 6 * p2
+        c = 3 * p1 - 3 * p0
+
+        disc = b * b - 4 * a * c
+
+        return Tuple.new unless disc >= 0
+
+        t1 = (-b + Math.sqrt(disc)) / (2 * a)
+        t2 = (-b - Math.sqrt(disc)) / (2 * a)
+
+        accept_1 = t1 >= 0 && t1 <= 1
+        accept_2 = t2 >= 0 && t2 <= 1
+
+        if accept_1 && accept_2
+          {t1, t2}
+        elsif accept_1
+          {t1}
+        elsif accept_2
+          {t2}
+        else
+          {0.5}
+        end
+      end
+
       property p0 : Vector2(T)
       property p1 : Vector2(T)
       property p2 : Vector2(T)
@@ -52,6 +78,17 @@ module PF
           T.new(-self.class.derivative(t, @p0.x, @p1.x, @p2.x, @p3.x)),
         ].normalized
       end
+
+      # Get the points at the extremeties of this curve
+      def extremeties
+        txs = self.class.extremeties(@p0.x, @p1.x, @p2.x, @p3.x)
+        tys = self.class.extremeties(@p0.y, @p1.y, @p2.y, @p3.y)
+
+        txs = txs.map { |t| at(t) }
+        tys = tys.map { |t| at(t) }
+
+        {txs, tys}
+      end
     end
   end
 end

src/sprite/draw_curve.cr

@@ -1,6 +1,6 @@
 module PF
   class Sprite
-    def draw_curve(curve : Bezier::Cubic, samples : Int = 100, pixel : Pixel = Pixel.new)
+    def draw_curve(curve : Bezier::Cubic | Bezier::Quad, samples : Int = 100, pixel : Pixel = Pixel.new)
       point = curve.p0
       0.upto(samples) do |x|
         t = x / samples
@@ -10,7 +10,7 @@ module PF
       end
     end
 
-    def draw_curve(curve : Bezier::Cubic, pixel : Pixel)
+    def draw_curve(curve : Bezier::Cubic | Bezier::Quad, pixel : Pixel)
       draw_curve(curve, pixel: pixel)
     end
   end