1 files changed, 119 insertions, 0 deletions
diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_torsion_intuition.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_torsion_intuition.py
new file mode 100644
index 0000000..31b9a85
--- /dev/null
+++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_torsion_intuition.py
@@ -0,0 +1,119 @@
+from manimlib.imports import *
+
+class t(SpecialThreeDScene):
+    CONFIG = {
+         "axes_config": {
+            "x_min": -5,
+            "x_max": 5,
+            "y_min": -5,
+            "y_max": 5,
+            "z_min": -4,
+            "z_max": 4,
+            "x_axis_config": {
+                "tick_frequency": 100,
+            },
+            "y_axis_config": {
+                "tick_frequency": 100,
+            },
+            "z_axis_config": {
+                "tick_frequency": 100,
+            },
+            "num_axis_pieces": 1,
+        }
+    }
+    def construct(self):
+
+        text = TextMobject(r'Torsion can be intuitively \\ thought of as the measure \\ of "twisting" of a curve.').scale(0.7).shift(2.5*UP + 4.2*LEFT)
+
+
+        dot = Dot().rotate(PI/2)
+        f1 = ParametricFunction(
+            lambda t: np.array([
+                2*np.sin(TAU*t),
+                2*np.cos(TAU*t),
+                2*t
+                ]), t_min = -2, t_max = 2, color = BLUE
+            ).scale(0.5)
+        d1 = Dot(color = RED).next_to(f1.get_center(), 2*DOWN + LEFT, buff = 0).shift(1.2*UP + 2.4*RIGHT)
+        t1 = self.get_torsion(2, 0.174)
+        t1 = "{:.2f}".format(t1)
+        t1 = TextMobject(fr'At the given point, $\tau = {t1}$').shift(3.5*DOWN).scale(0.7)
+
+        f2 = ParametricFunction(
+            lambda t: np.array([
+                3*np.sin(TAU*t),
+                3*np.cos(TAU*t),
+                2*t
+                ]), t_min = -2, t_max = 2, color = BLUE
+            ).scale(0.5)
+        d2 = Dot(color = RED).next_to(f2.get_center(), 2*DOWN + LEFT, buff = 0).shift(1.2*UP + 2.95*RIGHT)
+        t2 = self.get_torsion(3, 0.1765)
+        t2 = "{:.2f}".format(t2)
+        t2 = TextMobject(fr'At the given point, $\tau = {t2}$').shift(3.5*DOWN).scale(0.7)
+
+        f3 = ParametricFunction(
+            lambda t: np.array([
+                4*np.sin(TAU*t),
+                4*np.cos(TAU*t),
+                2*t
+                ]), t_min = -2, t_max = 2, color = BLUE
+            ).scale(0.5)
+        d3 = Dot(color = RED).next_to(f3.get_center(), 2*DOWN + LEFT, buff = 0).shift(1.2*UP + 3.45*RIGHT)
+        t3 = self.get_torsion(4, 0.179)
+        t3 = "{:.2f}".format(t3)
+        t3 = TextMobject(fr'At the given point, $\tau = {t3}$').shift(3.5*DOWN).scale(0.7)
+
+        f4 = ParametricFunction(
+            lambda t: np.array([
+                1.5*np.sin(TAU*t),
+                1.5*np.cos(TAU*t),
+                2*t
+                ]), t_min = -2, t_max = 2, color = BLUE
+            ).scale(0.5)
+        d4 = Dot(color = RED).next_to(f4.get_center(), 2*DOWN + LEFT, buff = 0).shift(1.215*UP + 2.128*RIGHT)
+        t4 = self.get_torsion(1.5, 0.173)
+        t4 = "{:.2f}".format(t4)
+        t4 = TextMobject(fr'At the given point, $\tau = {t4}$').shift(3.5*DOWN).scale(0.7)
+
+        f5 = ParametricFunction(
+            lambda t: np.array([
+                np.sin(TAU*t),
+                np.cos(TAU*t),
+                2*t
+                ]), t_min = -2, t_max = 2, color = BLUE
+            ).scale(0.5)
+
+        d5 = Dot(color = RED).next_to(f5.get_center(), 2*DOWN + LEFT, buff = 0).shift(1.3*UP + 1.858*RIGHT)
+        t5 = self.get_torsion(1, 0.17)
+        t5 = "{:.2f}".format(t5)
+        t5 = TextMobject(fr'At the given point, $\tau = {t5}$').shift(3.5*DOWN).scale(0.7)
+
+        axes = ThreeDAxes(**self.axes_config)
+        self.set_camera_orientation(phi = 60*DEGREES, theta=45*DEGREES)
+        self.add_fixed_in_frame_mobjects(t1, text)
+        self.play(FadeIn(VGroup(*[f1, d1, t1, axes, text])))
+        self.wait(2)
+        self.add_fixed_in_frame_mobjects(t2)
+        self.play(ReplacementTransform(d1, d2), ReplacementTransform(f1, f2), ReplacementTransform(t1, t2))
+        self.wait(2)
+        self.add_fixed_in_frame_mobjects(t3)
+        self.play(ReplacementTransform(d2, d3), ReplacementTransform(f2, f3), ReplacementTransform(t2, t3))
+        self.wait(2)
+        self.add_fixed_in_frame_mobjects(t4)
+        self.play(ReplacementTransform(d3, d4), ReplacementTransform(f3, f4), ReplacementTransform(t3, t4))
+        self.wait(2)
+        self.add_fixed_in_frame_mobjects(t5)
+        self.play(ReplacementTransform(d4, d5), ReplacementTransform(f4, f5), ReplacementTransform(t4, t5))
+        self.wait(2)
+        self.play(FadeOut(VGroup(*[d5, f5, t5, text, axes])))
+
+    def get_torsion(self, a, t):
+        rprime = np.array([a*np.cos(t), -a*np.sin(t), 2])
+        T = rprime / np.sqrt(np.dot(rprime, rprime))
+        rpp = np.array([-a*np.sin(t), -a*np.cos(t), 0])
+        n = rpp / np.dot(rpp, rpp)
+        b = np.cross(T, n)
+        dbdt = np.array([-2*np.sin(t), -2*np.cos(t), 0])
+        tor = np.dot(dbdt, n)
+
+        return tor