velocity updated, tnb torsion, fs1

5 files changed, 205 insertions, 46 deletions

diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.gif
deleted file mode 100644
index 6b4b438..0000000
--- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.gif
+++ /dev/null
diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.py
index c719a1d..f3f5a9c 100644
--- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.py
+++ b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file4_fs1.py
@@ -1,23 +1,91 @@
 from manimlib.imports import *
-
-class fs1(ThreeDScene):
+class fs1(GraphScene):
+    CONFIG = {
+        "x_min": -2,
+        "x_max": 2,
+        "y_min": -6,
+        "y_max": 6,
+        "graph_origin": ORIGIN
+    }
     def construct(self):
 
+        text = TextMobject(r'$\frac{dT}{ds} = \kappa N$ \\ $\frac{dT}{ds}$ gives the direction of N, \\ while $\kappa$ gives its magnitude.').scale(0.7).shift(3*UP + 3*LEFT)
+
+        self.setup_axes()
+        def curve_(x):
+            return x**3 - 2*x
+
+        def nm(x):
+            return abs(6 * x / ((9*(x**4) - 6*(x**2) + 5)**1.5))
+
+        figure = self.get_graph(curve_)
+
+
+        dot = Dot().rotate(PI/2)
+        alpha = ValueTracker(0)
+        t2_ = ValueTracker(-2)
+        t2 = t2_.get_value
+        t = alpha.get_value
+        vector_x = self.get_tangent_vector(t(),figure,scale=2)
+        vector_y = self.get_normal_vector(t(),figure,scale=2)
 
-        self.set_camera_orientation(phi = 75*DEGREES, theta=45*DEGREES)
-        dot1 = Dot(np.array([np.cos(-np.pi/3), np.sin(-np.pi/3), -0.4*np.pi/3]) + np.array([0,0.2,0]), radius = 0.16, color=RED)
-        tgt1 = Arrow((0,0,0), (-2,-0.55,0), color = YELLOW).shift(dot1.get_center() + np.array([0.18,0.04,0]))
-        nm1 = Arrow((0,0,0), (0.4,-2,0), color = BLUE).shift(dot1.get_center() + np.array([0,0.26,0])).shift(np.array([0.8,4.76,0])).rotate(-15*DEGREES).scale(0.8)
-        bnm1 = Arrow((0,0,0), (0,2,0), color=GREEN_E).shift(2.1*RIGHT+2*DOWN)
-        plane1 = Square(color = DARK_BROWN, fill_color = WHITE, fill_opacity=0.3).shift(dot1.get_center() + np.array([-0.4, -0.6, 0])).rotate(13*DEGREES).scale(1.2)
-        point1 = VGroup(*[dot1, tgt1, plane1]).scale(0.8).shift(np.array([1,4.86,0])).rotate(-15*DEGREES)
-        t = TextMobject(r'$T$', color = YELLOW).move_to(ORIGIN).shift(3.2*RIGHT + DOWN)
-        n = TextMobject(r'$N$', color = BLUE).shift(DOWN + RIGHT)
-        b = TextMobject(r'$B$', color = GREEN_E).next_to(bnm1, UP, buff = 0.1)
-        text = VGroup(*[t, n, b])
-        self.add_fixed_in_frame_mobjects(bnm1, text)
-        self.play(FadeIn(point1), FadeIn(bnm1), FadeIn(text))
-        self.wait()
-        self.play(TransformFromCopy(tgt1, nm1, run_time = 2))
+        kappa = TextMobject(r'$\kappa = $').scale(0.7).shift(3*DOWN + 3*RIGHT)
+
+        t_text = always_redraw(
+            lambda: DecimalNumber(
+                nm(t2()),
+                color=WHITE,
+            ).scale(0.7).next_to(kappa)
+        ).scale(0.6)
+
+        self.play(
+            ShowCreation(figure),
+            GrowFromCenter(dot),
+            GrowArrow(vector_x),
+            GrowArrow(vector_y)
+            )
+        vector_x.add_updater(
+            lambda m: m.become(
+                    self.get_tangent_vector(t(),figure,scale=2)
+                )
+            )
+        vector_y.add_updater(
+            lambda m: m.become(
+                    self.get_normal_vector(t(),figure,scale=2)
+                )
+            )
+        dot.add_updater(lambda m: m.move_to(vector_x.get_start()))
+        circle = Circle(radius = 2, color = GREEN_SCREEN). shift(2.63*RIGHT + 2.8*UP)
+        dot2 = Dot(np.array([2, curve_(2), 0]), color = WHITE).shift(2*DOWN + 2.5*RIGHT)
+
+        self.add(vector_x, vector_y,dot, t_text, kappa, text)
+        self.play(t2_.set_value, 2, alpha.set_value, 1, run_time=18, rate_func=smooth)
+        self.play(FadeIn(dot2), FadeIn(circle))
         self.wait(2)
-        self.play(FadeOut(VGroup(*[bnm1, text, point1, nm1])))
+        self.play(FadeOut(VGroup(*[self.axes, dot2, figure, circle, text, kappa, t_text])))
+
+
+    def get_tangent_vector(self, proportion, curve, dx=0.001, scale=0.5):
+        coord_i = curve.point_from_proportion(proportion)
+        coord_f = curve.point_from_proportion(proportion + dx)
+        reference_line = Line(coord_i,coord_f)
+        unit_vector = reference_line.get_unit_vector() * 0.7
+        vector = Arrow(coord_i , coord_i + unit_vector, color = YELLOW, buff=0)
+        return vector
+
+    def get_normal_vector(self, proportion, curve, dx=0.001, scale=1):
+        t = proportion.copy()/6
+        coord_i = curve.point_from_proportion(proportion)
+        coord_f = curve.point_from_proportion(proportion + dx)
+        length = 6 * t / ((9*(t**4) - 6*(t**2) + 5)**1.5)
+        if coord_i[0] <=  0 and coord_i[0] >  -0.5:
+            reference_line = Line(coord_i,coord_f).rotate(PI/2).set_width(0).scale(2)
+        elif coord_i[0] >  0 and (coord_i[0] < 0.5 or coord_i[0] > 2.7):
+            reference_line = Line(coord_i,coord_f).rotate(PI/2).set_width(0).scale(2)
+        elif coord_i[0] >  0:
+            reference_line = Line(coord_i,coord_f).rotate(PI/2).set_width(length).scale(2)
+        else:
+            reference_line = Line(coord_i,coord_f).rotate(-PI/2).set_width(length).scale(2)
+        unit_vector = reference_line.get_vector() * scale
+        vector = Arrow(coord_i , coord_i + unit_vector, color = RED_C, buff=0)
+        return vector
diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_fs2.gif b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_fs2.gif
deleted file mode 100644
index ce367b6..0000000
--- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_fs2.gif
+++ /dev/null
diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_fs2.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_fs2.py
deleted file mode 100644
index 0261fed..0000000
--- a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file5_fs2.py
+++ /dev/null
@@ -1,28 +0,0 @@
-from manimlib.imports import *
-
-class fs1(ThreeDScene):
-    def construct(self):
-
-
-        self.set_camera_orientation(phi = 75*DEGREES, theta=45*DEGREES)
-        dot1 = Dot(np.array([np.cos(-np.pi/3), np.sin(-np.pi/3), -0.4*np.pi/3]) + np.array([0,0.2,0]), radius = 0.16, color=RED)
-        tgt1 = Arrow((0,0,0), (-2,-0.55,0), color = YELLOW).shift(dot1.get_center() + np.array([0.18,0.04,0]))
-        nm1 = Arrow((0,0,0), (0.4,-2,0), color = BLUE).shift(dot1.get_center() + np.array([0,0.26,0])).shift(np.array([0.8,4.76,0])).rotate(-15*DEGREES).scale(0.8)
-        bnm1 = Arrow((0,0,0), (0,2,0), color=GREEN_E).shift(2.1*RIGHT+2*DOWN)
-
-        bnms = Line((0,0,0), (0,0,1.6), color = GREEN_E).shift(np.array([3.1,5.2,0])).scale(0.6)
-        bnmsa = ArrowTip(color = GREEN_E).next_to(bnms, np.array([0,0,1]), buff = 0).rotate(45*DEGREES)
-        bns = VGroup(*[bnms, bnmsa])
-
-        plane1 = Square(color = DARK_BROWN, fill_color = WHITE, fill_opacity=0.3).shift(dot1.get_center() + np.array([-0.4, -0.6, 0])).rotate(13*DEGREES).scale(1.2)
-        point1 = VGroup(*[dot1, tgt1, plane1]).scale(0.8).shift(np.array([1,4.86,0])).rotate(-15*DEGREES)
-        t = TextMobject(r'$T$', color = YELLOW).move_to(ORIGIN).shift(3.2*RIGHT + DOWN)
-        n = TextMobject(r'$N$', color = BLUE).shift(DOWN + RIGHT)
-        b = TextMobject(r'$B$', color = GREEN_E).next_to(bnm1, UP, buff = 0.1)
-        text = VGroup(*[t, n, b])
-        self.add_fixed_in_frame_mobjects(bnm1, text)
-        self.play(FadeIn(point1), FadeIn(text), FadeIn(bnm1))
-        self.wait()
-        self.play(TransformFromCopy(bnms, nm1, run_time = 3))
-        self.wait(2)
-        self.play(FadeOut(VGroup(*[bnms, text, point1, nm1, bnm1])))
diff --git a/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file6_torsion_intuition.py b/FSF-2020/calculus-of-several-variables/geometry-of-planes-and-curves/tnb-frame-and-serret-frenet-formulae/file6_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/file6_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