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图形学常见概念与算法-常用初等数学公式

作者:互联网

目录

乘法公式与因式分解

(a ± b)2 = a2 ± 2ab + b2

(a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc

a2 - b2 = (a - b)(a + b)

(a ± b)3 = a3 ± 3a2b + 3ab2 ± b3

a3 ± b3 = (a ± b)(a2 ∓ ab + b2)

an - bn = (a - b)(an-1 + an-2b + an-3b2 + … + abn-2+ bn-1)

比例 ( a b \frac{a}{b} ba​ = c d \frac{c}{d} dc​)

合比定理 a + b b \frac{a + b}{b} ba+b​ = c + d d \frac{c + d}{d} dc+d​

分比定理 a − b b \frac{a - b}{b} ba−b​ = c − d d \frac{c - d}{d} dc−d​

合分比定理 a + b a − b \frac{a + b}{a - b} a−ba+b​ = c + d c − d \frac{c + d}{c - d} c−dc+d​

二次方程 ax2 + bx + c = 0

根 = − b ± b 2 − 4 a c 2 a \frac{-b ± \sqrt{b^2 - 4ac}}{2a} 2a−b±b2−4ac ​​

韦达定理:x1 + x2 = - b a \frac{b}{a} ab​,x1x2 = c a \frac{c}{a} ac​

判别式△ = b2 - 4ac
△ > 0,方程有两个不等实根
△ = 0,方程有两个相等实根
△ < 0,方程有两个共轭虚根

对数

换底公式logaM = logbM / logba

loga1 = 0
logaa = 1

数列

等差数列,设a1为首项,an为通项,d为公差,Sn为n项和,则:

an = a1 + (n - 1)d
Sn = a   1   + a   n   2 \frac{a~1~ + a~n~}{2} 2a 1 +a n ​n
设a, b, c成等差数列,则等差中项b = 1 2 \frac{1}{2} 21​(a + c)

等比数列,设a1为首项,q为公比,an为通项,则:

通项an = a1qn-1
前n项和Sn = a 1   ( 1 − q n ) 1 − q \frac{a1~(1 - q^n)}{1 - q} 1−qa1 (1−qn)​ = a 1   − a n   q 1 − q \frac{a1~ - an~q}{1 - q} 1−qa1 −an q​

常用的几种数列的和

1 + 2 + 3 + … + n = 1 2 \frac{1}{2} 21​n(n - 1)

12 + 22 + 32 + … + n2 = 1 6 \frac{1}{6} 61​n(n + 1)(2n + 1)

13 + 23 + 33 + … + n3 = [ 1 2 \frac{1}{2} 21​n(n + 1)]2

1 * 2 + 2 * 3 + … + n(n+1) = 1 3 \frac{1}{3} 31​n(n + 1)(n + 2)

1 * 2 * 3 + 2 * 3 * 4 + … + n(n + 1)(n + 2) = 1 4 \frac{1}{4} 41​n(n + 1)(n + 2)(n + 3)

排列组合与二项式定理

排列:Pnm = n(n - 1)(n - 2)…[n - (m - 1)]

全排列:Pnn = n(n - 1)…3 * 2 * 1 = n!

组合:Cnm = n ( n − 1 ) . . . ( n − m + 1 ) m ! \frac{n(n - 1)...(n - m + 1)}{m!} m!n(n−1)...(n−m+1)​ = n ! m ! ( n − m ) ! \frac{n!}{m!(n - m)!} m!(n−m)!n!​

组合的性质:

Cnm = Cnn-m
Cnm = Cn-1m + Cn-1m-1

二项式定理:

(a + b)n = an + nan-1b + n ( n − 1 ) 2 ! \frac{n(n - 1)}{2!} 2!n(n−1)​an-2b2 + … + n ( n − 1 ) . . . ( n − ( k − 1 ) ) k ! \frac{n(n - 1)...(n - (k - 1))}{k!} k!n(n−1)...(n−(k−1))​an-kbk + … + bn

平面三角

三角函数间的关系:

sinα cscα = 1

cosα secα = 1

tanα cotα = 1

sin2α + cos2α = 1

1 + tan2α = sec2α

1 + cot2α = csc2α

tanα = s i n α c o s α \frac{sinα}{cosα} cosαsinα​

cotα = c o s α s i n α \frac{cosα}{sinα} sinαcosα​

倍角三角函数:

sin2α = 2 sinα cosα

cos2α = cos2α - sin2α = 1 - 2sin2α = 2cos2α - 1

tan2α = 2 t a n α 1 − t a n 2 α \frac{2tanα}{1 - tan^2α} 1−tan2α2tanα​

cot2α = 1 − c o t 2 α 2 c o t α \frac{1 - cot^2α}{2cotα} 2cotα1−cot2α​

sin2α = 1 − c o s 2 α 2 \frac{1 - cos2α}{2} 21−cos2α​

cos2α = 1 + c o s 2 α 2 \frac{1 + cos2α}{2} 21+cos2α​

三角函数的和差化积与积化和差公式:

sinα + sinβ = 2sin α + β 2 \frac{α + β}{2} 2α+β​cos α − β 2 \frac{α - β}{2} 2α−β​

sinα - sinβ = 2cos α + β 2 \frac{α + β}{2} 2α+β​sin α − β 2 \frac{α - β}{2} 2α−β​

cosα + cosβ = 2cos α + β 2 \frac{α + β}{2} 2α+β​cos α − β 2 \frac{α - β}{2} 2α−β​

cosα - cosβ = -2sin α + β 2 \frac{α + β}{2} 2α+β​sin α − β 2 \frac{α - β}{2} 2α−β​

sinα cosβ = 1 2 \frac{1}{2} 21​[sin(α + β) + sin(α - β)]

cosα cosβ = 1 2 \frac{1}{2} 21​[cos(α + β) + cos(α - β)]

cosα sinβ = 1 2 \frac{1}{2} 21​[sin(α + β) - sin(α - β)]

sinα sinβ = 1 2 \frac{1}{2} 21​[cos(α + β) - cos(α - β)]

正弦定理:

a s i n A \frac{a}{sinA} sinAa​ = b s i n B \frac{b}{sinB} sinBb​ = c s i n C \frac{c}{sinC} sinCc​ = 2R,R为外接圆半径

余弦定理:

a2 = b2 + c2 - 2bc cosA
b2 = c2 + a2 - 2ca cosB
c2 = a2 + b2 - 2ab cosC

反三角函数的恒等式:

arcsinx + arccosx = π 2 \frac{π}{2} 2π​

arctanx + arccotx = π 2 \frac{π}{2} 2π​

标签:cos,frac,21,图形学,算法,cos2,b2,初等数学,sin
来源: https://blog.csdn.net/u012043391/article/details/122785646