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Quintic Bezier Curves
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Expression 18: "f" Subscript, "l" "e" "r" "p" , Baseline left parenthesis, "P" Subscript, 1 , Baseline , "P" Subscript, 2 , Baseline , "t" , right parenthesis Has graph.
f
l
e
r
p
P
1
,
P
2
,
t
0
0
domain t Minimum:
less than or equal to "t" less than or equal to
≤
t
≤
1
1
domain t Maximum:
18
Expression 19: "f" Subscript, "l" "e" "r" "p" , Baseline left parenthesis, "P" Subscript, 2 , Baseline , "P" Subscript, 3 , Baseline , "t" , right parenthesis Has graph.
f
l
e
r
p
P
2
,
P
3
,
t
0
0
domain t Minimum:
less than or equal to "t" less than or equal to
≤
t
≤
1
1
domain t Maximum:
19
Expression 20: "T" equals 0.4 0 8
T
=
0
.
4
0
8
0
0
1
1
20
A: "A" equals "f" Subscript, "l" "e" "r" "p" , Baseline left parenthesis, "P" Subscript, 0 , Baseline , "P" Subscript, 1 , Baseline , "T" , right parenthesis
A
=
f
l
e
r
p
P
0
,
P
1
,
T
Label:
equals
=
left parenthesis, negative 0.2 8 6 9 4 1 , 0.5 2 3 3 3 , right parenthesis
−
0
.
2
8
6
9
4
1
,
0
.
5
2
3
3
3
21
B: "B" equals "f" Subscript, "l" "e" "r" "p" , Baseline left parenthesis, "P" Subscript, 1 , Baseline , "P" Subscript, 2 , Baseline , "T" , right parenthesis
B
=
f
l
e
r
p
P
1
,
P
2
,
T
Label:
equals
=
left parenthesis, 1.0 4 8 7 1 2 , negative 3.0 7 3 9 1 , right parenthesis
1
.
0
4
8
7
1
2
,
−
3
.
0
7
3
9
1
22
C: "C" equals "f" Subscript, "l" "e" "r" "p" , Baseline left parenthesis, "P" Subscript, 2 , Baseline , "P" Subscript, 3 , Baseline , "T" , right parenthesis
C
=
f
l
e
r
p
P
2
,
P
3
,
T
Label:
equals
=
left parenthesis, 0.7 1 6 3 9 , negative 3.5 0 0 3 2 , right parenthesis
0
.
7
1
6
3
9
,
−
3
.
5
0
0
3
2
23
Expression 24: "f" Subscript, "l" "e" "r" "p" , Baseline left parenthesis, "A" , "B" , "t" , right parenthesis
f
l
e
r
p
A
,
B
,
t
0
0
domain t Minimum:
less than or equal to "t" less than or equal to
≤
t
≤
1
1
domain t Maximum:
24
Expression 25: "f" Subscript, "l" "e" "r" "p" , Baseline left parenthesis, "A" , "B" , "t" , right parenthesis
f
l
e
r
p
A
,
B
,
t
0
0
domain t Minimum:
less than or equal to "t" less than or equal to
≤
t
≤
1
1
domain t Maximum:
25
Expression 26: "f" Subscript, "l" "e" "r" "p" , Baseline left parenthesis, "B" , "C" , "t" , right parenthesis
f
l
e
r
p
B
,
C
,
t
0
0
domain t Minimum:
less than or equal to "t" less than or equal to
≤
t
≤
1
1
domain t Maximum:
26
D: "D" equals "f" Subscript, "l" "e" "r" "p" , Baseline left parenthesis, "A" , "B" , "T" , right parenthesis
D
=
f
l
e
r
p
A
,
B
,
T
Label:
equals
=
left parenthesis, 0.2 2 4 6 1 4 1 , negative 0.8 5 4 4 1 2 9 2 , right parenthesis
0
.
2
2
4
6
1
4
1
,
−
0
.
8
5
4
4
1
2
9
2
27
E: "E" equals "f" Subscript, "l" "e" "r" "p" , Baseline left parenthesis, "B" , "C" , "T" , right parenthesis
E
=
f
l
e
r
p
B
,
C
,
T
Label:
equals
=
left parenthesis, 0.9 2 1 4 3 2 6 7 , negative 3.2 3 7 2 2 5 0 3 , right parenthesis
0
.
9
2
1
4
3
2
6
7
,
−
3
.
2
3
7
2
2
5
0
3
28
Expression 29: "f" Subscript, "l" "e" "r" "p" , Baseline left parenthesis, "D" , "E" , "t" , right parenthesis
f
l
e
r
p
D
,
E
,
t
0
0
domain t Minimum:
less than or equal to "t" less than or equal to
≤
t
≤
domain t Maximum: 1
1
29
P: "P" equals "f" Subscript, "l" "e" "r" "p" , Baseline left parenthesis, "D" , "E" , "T" , right parenthesis
P
=
f
l
e
r
p
D
,
E
,
T
Label:
equals
=
left parenthesis, 0.4 9 1 4 9 5 6 1 , negative 1.7 6 7 0 2 9 9 6 , right parenthesis
0
.
4
9
1
4
9
5
6
1
,
−
1
.
7
6
7
0
2
9
9
6
30
Expression 31: "f" Subscript, "B" "e" "r" "n" "s" "t" "e" "i" "n" "P" "o" "l" "y" "n" "o" "m" "i" "a" "l" , Baseline left parenthesis, "P" Subscript, 0 , Baseline , "P" Subscript, 1 , Baseline , "P" Subscript, 2 , Baseline , "P" Subscript, 3 , Baseline , "t" , right parenthesis
f
B
e
r
n
s
t
e
i
n
P
o
l
y
n
o
m
i
a
l
P
0
,
P
1
,
P
2
,
P
3
,
t
0
0
domain t Minimum:
less than or equal to "t" less than or equal to
≤
t
≤
1
1
domain t Maximum:
31
Functions
Functions
Hide this folder from students.
32
LERP = linear interpolation
LERP = linear interpolation
33
Expression 34: "f" Subscript, "l" "e" "r" "p" , Baseline left parenthesis, "P" Subscript, "a" , Baseline , "P" Subscript, "b" , Baseline , "t" , right parenthesis equals left parenthesis, 1 minus "t" , right parenthesis "P" Subscript, "a" , Baseline plus "t" "P" Subscript, "b" , Baseline
f
l
e
r
p
P
a
,
P
b
,
t
=
1
−
t
P
a
+
t
P
b
34
Expression 35: "f" Subscript, "B" "e" "r" "n" "s" "t" "e" "i" "n" "P" "o" "l" "y" "n" "o" "m" "i" "a" "l" , Baseline left parenthesis, "P" Subscript, "a" , Baseline , "P" Subscript, "b" , Baseline , "P" Subscript, "c" , Baseline , "P" Subscript, "d" , Baseline , "t" , right parenthesis equals left parenthesis, negative "P" Subscript, "a" , Baseline "t" cubed plus 3 "P" Subscript, "a" , Baseline "t" squared minus 3 "P" Subscript, "a" , Baseline "t" plus "P" Subscript, "a" , Baseline , right parenthesis plus left parenthesis, 3 "P" Subscript, "b" , Baseline "t" cubed minus 6 "P" Subscript, "b" , Baseline "t" squared plus 3 "P" Subscript, "b" , Baseline "t" , right parenthesis plus left parenthesis, negative 3 "P" Subscript, "c" , Baseline "t" cubed plus 3 "P" Subscript, "c" , Baseline "t" squared , right parenthesis plus left parenthesis, "P" Subscript, "d" , Baseline , right parenthesis "t" cubed
f
B
e
r
n
s
t
e
i
n
P
o
l
y
n
o
m
i
a
l
P
a
,
P
b
,
P
c
,
P
d
,
t
=
−
P
a
t
3
+
3
P
a
t
2
−
3
P
a
t
+
P
a
+
3
P
b
t
3
−
6
P
b
t
2
+
3
P
b
t
+
−
3
P
c
t
3
+
3
P
c
t
2
+
P
d
t
3
35
Expression 36:
36
37
powered by
powered by
"P" Subscript, 0 , Baseline
P
0
"P" Subscript, 1 , Baseline
P
1
"P" Subscript, 2 , Baseline
P
2
"P" Subscript, 3 , Baseline
P
3
A
B
C
D
E
P
"x"
x
"y"
y
"a" squared
a
2
"a" Superscript, "b" , Baseline
a
b
7
7
8
8
9
9
over
÷
functions
(
(
)
)
less than
<
greater than
>
4
4
5
5
6
6
times
×
| "a" |
|
a
|
,
,
less than or equal to
≤
greater than or equal to
≥
1
1
2
2
3
3
negative
−
A B C
StartRoot, , EndRoot
pi
π
0
0
.
.
equals
=
positive
+
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