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Computer Graphics
Common sets of interest include
• R—the real numbers;
• R+—the nonnegative real numbers (includes zero);
• R2—the ordered pairs in the real 2D plane;
• Rn—the points in n-dimensional Cartesian space;
• Z—the integers;
• S2—the set of 3D points (points in R3) on the unit sphere.
Logarithms
The “log base a” of x is written
l
o
g
a
x
log_a{x}
logax and is defined as “the exponent to which a must be raised to get x.”
Note that the logarithm base a and the function that raises a to a power are inverses of each other.
y
=
log
a
x
⇔
a
y
=
x
y = \log_a{x} ⇔ a^y = x
y=logax⇔ay=x
a
log
a
(
x
)
=
x
a^{\log_a{(x)}} = x
aloga(x)=x
log
a
(
a
x
)
=
x
\log_a({a^x}) = x
loga(ax)=x
log
a
(
x
y
)
=
log
a
x
+
log
a
y
\log_a{(xy)} = \log_a{x} + \log_a{y}
loga(xy)=logax+logay
log
a
(
x
/
y
)
=
log
a
x
−
log
a
y
\log_a{(x/y)} = \log_a{x} - \log_a{y}
loga(x/y)=logax−logay
log
a
x
=
log
a
b
∗
log
a
x
\log_a{x} = \log_a{b} * \log_a{x}
logax=logab∗logax
The logarithm with base e is called the natural logarithm.
ln
x
≡
log
e
y
\ln{x} \equiv \log_e{y}
lnx≡logey
Solving Quadratic Equations
A quadratic equation has the form:$ Ax^2+Bx+C=0 $
x
=
−
B
±
B
2
−
4
A
C
2
A
x = \frac{-B\pm \sqrt{B^2 - 4AC}}{2A}
x=2A−B±B2−4AC
D
≡
B
2
−
4
A
C
D \equiv B^2 - 4AC
D≡B2−4AC
Trigonometry
d
e
g
r
e
e
s
=
180
π
r
a
d
i
a
n
s
degrees = \frac{180}{\pi}radians
degrees=π180radians
r
a
d
i
a
n
s
=
π
180
d
e
g
r
e
e
s
radians = \frac{\pi}{180}degrees
radians=180πdegrees
Shifting identities:
sin
(
−
A
)
=
−
sin
(
A
)
\sin(-A) = -\sin(A)
sin(−A)=−sin(A)
cos
(
−
A
)
=
cos
(
A
)
\cos(-A) = \cos(A)
cos(−A)=cos(A)
tan
(
−
A
)
=
−
tan
(
A
)
\tan(-A) = -\tan(A)
tan(−A)=−tan(A)
sin
(
π
2
−
A
)
=
cos
(
A
)
\sin(\frac{\pi}{2}-A) = \cos(A)
sin(2π−A)=cos(A)
cos
(
π
2
−
A
)
=
sin
(
A
)
\cos(\frac{\pi}{2}-A) = \sin(A)
cos(2π−A)=sin(A)
tan
(
π
2
−
A
)
=
cot
(
A
)
\tan(\frac{\pi}{2}-A) = \cot(A)
tan(2π−A)=cot(A)
Pythagorean identities:
sin
2
A
+
cos
2
A
=
1
\sin^2A + \cos^2A=1
sin2A+cos2A=1
sec
2
A
−
tan
2
A
=
1
(
s
e
c
=
c
o
s
−
1
)
\sec^2A - \tan^2A=1 (sec = cos^{-1})
sec2A−tan2A=1(sec=cos−1)
csc
2
A
−
cot
2
A
=
1
(
c
s
c
=
s
i
n
−
1
)
\csc^2A - \cot^2A=1 (csc= sin^{-1})
csc2A−cot2A=1(csc=sin−1)
Addition and subtraction identities:
sin
(
A
+
B
)
=
sin
A
cos
B
+
sin
B
cos
A
\sin(A+B) = \sin{A}\cos{B}+\sin{B}\cos{A}
sin(A+B)=sinAcosB+sinBcosA
sin
(
A
−
B
)
=
sin
A
cos
B
−
sin
B
cos
A
\sin(A-B) = \sin{A}\cos{B}-\sin{B}\cos{A}
sin(A−B)=sinAcosB−sinBcosA
sin
(
2
A
)
=
2
sin
A
cos
B
\sin(2A) = 2\sin{A}\cos{B}
sin(2A)=2sinAcosB
本文标签： GraphicscomputerfundamentalNoteMath
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