feat: adding three new rate_functions based on the SmoothStep function. These have the added benefits of zero derivatives at the endpoints. (#3361)

Author: uwezi

manim/utils/rate_functions.py

@@ -89,6 +89,9 @@ def construct(self):
 __all__ = [
     "linear",
     "smooth",
+    "smoothstep",
+    "smootherstep",
+    "smoothererstep",
     "rush_into",
     "rush_from",
     "slow_into",
@@ -155,6 +158,36 @@ def smooth(t: float, inflection: float = 10.0) -> float:
     )
 
 
+def smoothstep(t: float) -> float:
+    """Implementation of the 1st order SmoothStep sigmoid function.
+    The 1st derivative (speed) is zero at the endpoints.
+    https://en.wikipedia.org/wiki/Smoothstep
+    """
+    return 0 if t <= 0 else 3 * t**2 - 2 * t**3 if t < 1 else 1
+
+
+def smootherstep(t: float) -> float:
+    """Implementation of the 2nd order SmoothStep sigmoid function.
+    The 1st and 2nd derivatives (speed and acceleration) are zero at the endpoints.
+    https://en.wikipedia.org/wiki/Smoothstep
+    """
+    return 0 if t <= 0 else 6 * t**5 - 15 * t**4 + 10 * t**3 if t < 1 else 1
+
+
+def smoothererstep(t: float) -> float:
+    """Implementation of the 3rd order SmoothStep sigmoid function.
+    The 1st, 2nd and 3rd derivatives (speed, acceleration and jerk) are zero at the endpoints.
+    https://en.wikipedia.org/wiki/Smoothstep
+    """
+    return (
+        0
+        if t <= 0
+        else 35 * t**4 - 84 * t**5 + 70 * t**6 - 20 * t**7
+        if t < 1
+        else 1
+    )
+
+
 @unit_interval
 def rush_into(t: float, inflection: float = 10.0) -> float:
     return 2 * smooth(t / 2.0, inflection)