def get_det(self):
matrix = Matrix(np.array([
["\\hat{\\imath}", "\\hat{\\jmath}", "\\hat{k}"],
["v_%d"%d for d in range(1, 4)],
["w_%d"%d for d in range(1, 4)],
]).T)
matrix.highlight_columns(X_COLOR, V_COLOR, W_COLOR)
matrix.get_mob_matrix()[1, 0].highlight(Y_COLOR)
matrix.get_mob_matrix()[2, 0].highlight(Z_COLOR)
Group(*matrix.get_mob_matrix()[1, 1:]).shift(0.15*DOWN)
Group(*matrix.get_mob_matrix()[2, 1:]).shift(0.35*DOWN)
det_text = get_det_text(matrix)
det_text.add(matrix)
return det_text
def get_det(self):
matrix = Matrix(
np.array([
["\\hat{\\imath}", "\\hat{\\jmath}", "\\hat{k}"],
["v_%d" % d for d in range(1, 4)],
["w_%d" % d for d in range(1, 4)],
]).T)
matrix.highlight_columns(X_COLOR, V_COLOR, W_COLOR)
matrix.get_mob_matrix()[1, 0].highlight(Y_COLOR)
matrix.get_mob_matrix()[2, 0].highlight(Z_COLOR)
VGroup(*matrix.get_mob_matrix()[1, 1:]).shift(0.15 * DOWN)
VGroup(*matrix.get_mob_matrix()[2, 1:]).shift(0.35 * DOWN)
det_text = get_det_text(matrix)
det_text.add(matrix)
return det_text
def construct(self):
v_tex, w_tex, p_tex = get_vect_tex(*"vwp")
vector_word = TextMobject("Vector:")
transform_word = TextMobject("Dual transform:")
cross = TexMobject(p_tex, "=", v_tex, "\\times", w_tex)
for tex, color in zip([v_tex, w_tex, p_tex],
[U_COLOR, W_COLOR, P_COLOR]):
cross.highlight_by_tex(tex, color)
input_array_tex = matrix_to_tex_string(["x", "y", "z"])
func = TexMobject("L\\left(%s\\right) = " % input_array_tex)
matrix = Matrix(
np.array([
["x", "y", "z"],
["v_1", "v_2", "v_3"],
["w_1", "w_2", "w_3"],
]).T)
matrix.highlight_columns(WHITE, U_COLOR, W_COLOR)
det_text = get_det_text(matrix, background_rect=False)
det_text.add(matrix)
dot_with_cross = TexMobject("%s \\cdot ( " % input_array_tex, v_tex,
"\\times", w_tex, ")")
dot_with_cross.highlight_by_tex(v_tex, U_COLOR)
dot_with_cross.highlight_by_tex(w_tex, W_COLOR)
transform = Group(func, det_text)
transform.arrange_submobjects()
Group(transform, dot_with_cross).scale(0.7)
Group(vector_word, cross).arrange_submobjects(
RIGHT, buff=MED_BUFF).center().shift(LEFT).to_edge(UP)
transform_word.next_to(vector_word,
DOWN,
buff=MED_BUFF,
aligned_edge=LEFT)
transform.next_to(transform_word,
DOWN,
buff=MED_BUFF,
aligned_edge=LEFT)
dot_with_cross.next_to(func, RIGHT)
self.add(vector_word)
self.play(Write(cross))
self.dither()
self.play(FadeIn(transform_word))
self.play(Write(transform))
self.dither()
self.play(Transform(det_text, dot_with_cross))
self.dither()
def show_triple_cross_product(self):
colors = [WHITE, ORANGE, W_COLOR]
tex_mobs = map(TexMobject, get_vect_tex(*"uvw"))
u_tex, v_tex, w_tex = tex_mobs
arrays = [
Matrix(["%s_%d" % (s, d) for d in range(1, 4)]) for s in "uvw"
]
defs_equals = VGroup()
definitions = VGroup()
for array, tex_mob, color in zip(arrays, tex_mobs, colors):
array.highlight_columns(color)
tex_mob.highlight(color)
equals = TexMobject("=")
definition = VGroup(tex_mob, equals, array)
definition.arrange_submobjects(RIGHT)
definitions.add(definition)
defs_equals.add(equals)
definitions.arrange_submobjects(buff=MED_SMALL_BUFF)
definitions.shift(2 * DOWN)
mobs_with_targets = list(
it.chain(tex_mobs, *[a.get_entries() for a in arrays]))
for mob in mobs_with_targets:
mob.target = mob.copy()
matrix = Matrix(
np.array([[e.target for e in array.get_entries()]
for array in arrays]).T)
det_text = get_det_text(matrix, background_rect=False)
syms = times1, times2, equals = [
TexMobject(sym) for sym in "\\times", "\\times", "=",
]
triple_cross = VGroup(u_tex.target, times1, v_tex.target, times2,
w_tex.target, equals)
triple_cross.arrange_submobjects()
final_mobs = VGroup(triple_cross, VGroup(det_text, matrix))
final_mobs.arrange_submobjects()
final_mobs.next_to(self.title, DOWN, buff=MED_SMALL_BUFF)
for mob in definitions, final_mobs:
mob.scale_to_fit_width(SPACE_WIDTH - 1)
for array in arrays:
brackets = array.get_brackets()
brackets.target = matrix.get_brackets()
mobs_with_targets.append(brackets)
for def_equals in defs_equals:
def_equals.target = equals
mobs_with_targets.append(def_equals)
self.play(
FadeIn(definitions, run_time=2, submobject_mode="lagged_start"))
self.dither(2)
self.play(*[Transform(mob.copy(), mob.target) for mob in tex_mobs] + [
Write(times1),
Write(times2),
])
triple_cross.add(*self.get_mobjects_from_last_animation()[:3])
self.play(*[
Transform(mob.copy(), mob.target) for mob in mobs_with_targets
if mob not in tex_mobs
])
u_entries = self.get_mobjects_from_last_animation()[:3]
v_entries = self.get_mobjects_from_last_animation()[3:6]
w_entries = self.get_mobjects_from_last_animation()[6:9]
self.play(Write(det_text))
self.dither(2)
self.det_text = det_text
self.definitions = definitions
self.u_entries = u_entries
self.v_entries = v_entries
self.w_entries = w_entries
self.matrix = matrix
self.triple_cross = triple_cross
self.v_tex, self.w_tex = v_tex, w_tex
self.equals = equals
def show_triple_cross_product(self):
colors = [WHITE, ORANGE, W_COLOR]
tex_mobs = map(TexMobject, get_vect_tex(*"uvw"))
u_tex, v_tex, w_tex = tex_mobs
arrays = [
Matrix(["%s_%d"%(s, d) for d in range(1, 4)])
for s in "uvw"
]
defs_equals = Group()
definitions = Group()
for array, tex_mob, color in zip(arrays, tex_mobs, colors):
array.highlight_columns(color)
tex_mob.highlight(color)
equals = TexMobject("=")
definition = Group(tex_mob, equals, array)
definition.arrange_submobjects(RIGHT)
definitions.add(definition)
defs_equals.add(equals)
definitions.arrange_submobjects(buff = MED_BUFF)
definitions.shift(2*DOWN)
mobs_with_targets = list(it.chain(
tex_mobs, *[a.get_entries() for a in arrays]
))
for mob in mobs_with_targets:
mob.target = mob.copy()
matrix = Matrix(np.array([
[e.target for e in array.get_entries()]
for array in arrays
]).T)
det_text = get_det_text(matrix, background_rect = False)
syms = times1, times2, equals = [
TexMobject(sym)
for sym in "\\times", "\\times", "=",
]
triple_cross = Group(
u_tex.target, times1, v_tex.target, times2, w_tex.target, equals
)
triple_cross.arrange_submobjects()
final_mobs = Group(triple_cross, Group(det_text, matrix))
final_mobs.arrange_submobjects()
final_mobs.next_to(self.title, DOWN, buff = MED_BUFF)
for mob in definitions, final_mobs:
mob.scale_to_fit_width(SPACE_WIDTH - 1)
for array in arrays:
brackets = array.get_brackets()
brackets.target = matrix.get_brackets()
mobs_with_targets.append(brackets)
for def_equals in defs_equals:
def_equals.target = equals
mobs_with_targets.append(def_equals)
self.play(FadeIn(
definitions,
run_time = 2,
submobject_mode = "lagged_start"
))
self.dither(2)
self.play(*[
Transform(mob.copy(), mob.target)
for mob in tex_mobs
] + [
Write(times1),
Write(times2),
])
triple_cross.add(*self.get_mobjects_from_last_animation()[:3])
self.play(*[
Transform(mob.copy(), mob.target)
for mob in mobs_with_targets
if mob not in tex_mobs
])
u_entries = self.get_mobjects_from_last_animation()[:3]
v_entries = self.get_mobjects_from_last_animation()[3:6]
w_entries = self.get_mobjects_from_last_animation()[6:9]
self.play(Write(det_text))
self.dither(2)
self.det_text = det_text
self.definitions = definitions
self.u_entries = u_entries
self.v_entries = v_entries
self.w_entries = w_entries
self.matrix = matrix
self.triple_cross = triple_cross
self.v_tex, self.w_tex = v_tex, w_tex
self.equals = equals
def construct(self):
self.plane.fade()
v = Vector(self.v_coords, color=V_COLOR)
w = Vector(self.w_coords, color=W_COLOR)
v.coords = Matrix(self.v_coords)
w.coords = Matrix(self.w_coords)
v.coords.next_to(v.get_end(), LEFT)
w.coords.next_to(w.get_end(), RIGHT)
v.coords.highlight(v.get_color())
w.coords.highlight(w.get_color())
for coords in v.coords, w.coords:
coords.background_rectangle = BackgroundRectangle(coords)
coords.add_to_back(coords.background_rectangle)
v.label = self.get_vector_label(v, "v", "left", color=v.get_color())
w.label = self.get_vector_label(w, "w", "right", color=w.get_color())
matrix = Matrix(
np.array([
list(v.coords.copy().get_entries()),
list(w.coords.copy().get_entries()),
]).T)
matrix_background = BackgroundRectangle(matrix)
col1, col2 = it.starmap(Group, matrix.get_mob_matrix().T)
det_text = get_det_text(matrix)
v_tex, w_tex = get_vect_tex("v", "w")
cross_product = TexMobject(v_tex, "\\times", w_tex, "=")
cross_product.highlight_by_tex(v_tex, V_COLOR)
cross_product.highlight_by_tex(w_tex, W_COLOR)
cross_product.add_background_rectangle()
equation_start = Group(cross_product,
Group(matrix_background, det_text, matrix))
equation_start.arrange_submobjects()
equation_start.next_to(ORIGIN, DOWN).to_edge(LEFT)
for vect in v, w:
self.play(ShowCreation(vect), Write(vect.coords),
Write(vect.label))
self.dither()
self.play(
Transform(v.coords.background_rectangle, matrix_background),
Transform(w.coords.background_rectangle, matrix_background),
Transform(v.coords.get_entries(), col1),
Transform(w.coords.get_entries(), col2),
Transform(v.coords.get_brackets(), matrix.get_brackets()),
Transform(w.coords.get_brackets(), matrix.get_brackets()),
)
self.play(*map(Write, [det_text, cross_product]))
v1, v2 = v.coords.get_entries()
w1, w2 = w.coords.get_entries()
entries = v1, v2, w1, w2
for entry in entries:
entry.target = entry.copy()
det = np.linalg.det([self.v_coords, self.w_coords])
equals, dot1, minus, dot2, equals_result = syms = Group(
*map(TexMobject, ["=", "\\cdot", "-", "\\cdot",
"=%d" % det]))
equation_end = Group(equals, v1.target, dot1, w2.target, minus,
w1.target, dot2, v2.target, equals_result)
equation_end.arrange_submobjects()
equation_end.next_to(equation_start)
syms_rect = BackgroundRectangle(syms)
syms.add_to_back(syms_rect)
equation_end.add_to_back(syms_rect)
syms.remove(equals_result)
self.play(Write(syms),
Transform(Group(v1, w2).copy(),
Group(v1.target, w2.target),
rate_func=squish_rate_func(smooth, 0, 1. / 3),
path_arc=np.pi / 2),
Transform(Group(v2, w1).copy(),
Group(v2.target, w1.target),
rate_func=squish_rate_func(smooth, 2. / 3, 1),
path_arc=np.pi / 2),
run_time=3)
self.dither()
self.play(Write(equals_result))
self.add_foreground_mobject(equation_start, equation_end)
self.show_transformation(v, w)
det_sym = TexMobject(str(int(abs(det))))
det_sym.scale(1.5)
det_sym.next_to(v.get_end() + w.get_end(),
DOWN + RIGHT,
buff=MED_BUFF / 2)
arc = self.get_arc(v, w, radius=1)
arc.highlight(RED)
self.play(Write(det_sym))
self.play(ShowCreation(arc))
self.dither()
def initial_definition(self):
self.plane.save_state()
self.plane.fade()
v = self.add_vector(self.v_coords, color=V_COLOR)
w = self.add_vector(self.w_coords, color=W_COLOR)
self.moving_vectors.remove(v)
self.moving_vectors.remove(w)
for vect, name, direction in (v, "v", "left"), (w, "w", "right"):
color = vect.get_color()
vect.label = self.label_vector(
vect,
name,
color=color,
direction=direction,
)
vect.coord_array = vector_coordinate_label(
vect,
color=color,
)
vect.coords = vect.coord_array.get_entries()
for vect, edge in (v, DOWN), (w, UP):
vect.coord_array.move_to(vect.coord_array.get_center(),
aligned_edge=edge)
self.play(Write(vect.coord_array, run_time=1))
movers = [v.label, w.label, v.coords, w.coords]
for mover in movers:
mover.target = mover.copy()
times = TexMobject("\\times")
cross_product = Group(v.label.target, times, w.label.target)
cross_product.arrange_submobjects()
matrix = Matrix(
np.array([list(v.coords.target),
list(w.coords.target)]).T)
det_text = get_det_text(matrix)
full_det = Group(det_text, matrix)
equals = TexMobject("=")
equation = Group(cross_product, equals, full_det)
equation.arrange_submobjects()
equation.to_corner(UP + LEFT)
matrix_background = BackgroundRectangle(matrix)
cross_background = BackgroundRectangle(cross_product)
disclaimer = TextMobject(
"$^*$ See ``Note on conventions'' in description")
disclaimer.scale(0.7)
disclaimer.highlight(RED)
disclaimer.next_to(det_text.get_corner(UP + RIGHT), RIGHT, buff=0)
disclaimer.add_background_rectangle()
self.play(
FadeIn(cross_background),
Transform(v.label.copy(), v.label.target),
Transform(w.label.copy(), w.label.target),
Write(times),
)
self.dither()
self.play(ShowCreation(matrix_background),
Write(matrix.get_brackets()),
run_time=1)
self.play(Transform(v.coords.copy(), v.coords.target))
self.play(Transform(w.coords.copy(), w.coords.target))
matrix.add_to_back(matrix_background)
self.dither()
self.play(
Write(equals),
Write(det_text),
Animation(matrix),
)
self.dither()
self.play(FadeIn(disclaimer))
self.dither()
self.play(FadeOut(disclaimer))
self.dither()
cross_product.add_to_back(cross_background)
cross_product.add(equals)
self.cross_product = cross_product
self.matrix = matrix
self.det_text = det_text
self.v, self.w = v, w
def construct(self):
v_terms, w_terms = [["%s_%d" % (s, d) for d in range(1, 4)]
for s in "v", "w"]
v = Matrix(v_terms)
w = Matrix(w_terms)
v.highlight(V_COLOR)
w.highlight(W_COLOR)
matrix = Matrix(
np.array([
[
TexMobject("\\hat{%s}" % s)
for s in "\\imath", "\\jmath", "k"
],
list(v.get_entries().copy()),
list(w.get_entries().copy()),
]).T)
colors = [X_COLOR, Y_COLOR, Z_COLOR]
col1, col2, col3 = it.starmap(Group, matrix.get_mob_matrix().T)
i, j, k = col1
v1, v2, v3 = col2
w1, w2, w3 = col3
##Really should fix Matrix mobject...
j.shift(0.1 * UP)
k.shift(0.2 * UP)
Group(v2, w2).shift(0.1 * DOWN)
Group(v3, w3).shift(0.2 * DOWN)
##
for color, entry in zip(colors, col1):
entry.highlight(color)
det_text = get_det_text(matrix)
equals = TexMobject("=")
equation = Group(v, TexMobject("\\times"), w, equals,
Group(det_text, matrix))
equation.arrange_submobjects()
self.add(*equation[:-2])
self.dither()
self.play(Write(matrix.get_brackets()))
for col, vect in (col2, v), (col3, w):
col.save_state()
col.move_to(vect.get_entries())
self.play(
col.restore,
path_arc=-np.pi / 2,
)
for entry in col1:
self.play(Write(entry))
self.dither()
self.play(*map(Write, [equals, det_text]))
self.dither()
disclaimer = TextMobject(
"$^*$ See ``Note on conventions'' in description")
disclaimer.scale(0.7)
disclaimer.highlight(RED)
disclaimer.next_to(equation, DOWN)
self.play(FadeIn(disclaimer))
self.dither()
self.play(FadeOut(disclaimer))
circle = Circle()
circle.stretch_to_fit_height(col1.get_height() + 1)
circle.stretch_to_fit_width(col1.get_width() + 1)
circle.move_to(col1)
randy = Randolph()
randy.scale(0.9)
randy.to_corner()
randy.to_edge(DOWN, buff=SMALL_BUFF)
self.play(FadeIn(randy))
self.play(randy.change_mode, "confused", ShowCreation(circle))
self.play(randy.look, RIGHT)
self.dither()
self.play(FadeOut(circle))
self.play(
equation.to_corner, UP + LEFT,
ApplyFunction(lambda r: r.change_mode("plain").look(UP + RIGHT),
randy))
quints = [
(i, v2, w3, v3, w2),
(j, v3, w1, v1, w3),
(k, v1, w2, v2, w1),
]
last_mob = None
paren_sets = []
for quint in quints:
for mob in quint:
mob.t = mob.copy()
mob.save_state()
basis = quint[0]
basis.t.scale(1 / 0.8)
lp, minus, rp = syms = Group(*map(TexMobject, "(-)"))
term = Group(basis.t, lp, quint[1].t, quint[2].t, minus,
quint[3].t, quint[4].t, rp)
term.arrange_submobjects()
if last_mob:
plus = TexMobject("+")
syms.add(plus)
plus.next_to(term, LEFT, buff=MED_BUFF / 2)
term.add_to_back(plus)
term.next_to(last_mob, RIGHT, buff=MED_BUFF / 2)
else:
term.next_to(equation, DOWN, buff=MED_BUFF, aligned_edge=LEFT)
last_mob = term
self.play(*it.chain(*[[mob.scale_in_place, 1.2] for mob in quint]))
self.dither()
self.play(*[Transform(mob.copy(), mob.t)
for mob in quint] + [mob.restore
for mob in quint] + [Write(syms)],
run_time=2)
self.dither()
paren_sets.append(Group(lp, rp))
self.dither()
self.play(randy.change_mode, "pondering")
for parens in paren_sets:
brace = Brace(parens)
text = brace.get_text("Some number")
text.scale_to_fit_width(brace.get_width())
self.play(GrowFromCenter(brace), Write(text, run_time=2))
self.dither()