1、Geometry幾何對象原著Ed AngelProfessor of Computer Science, Electrical and Computer Engineering, and Media ArtsUniversity of New Mexico編輯 武漢大學(xué)計算機(jī)學(xué)院圖形學(xué)課程組Objectives(對象)Introduce the elements of geometry幾何要素Scalars標(biāo)量Vectors向量Points點Develop mathematical operations among them in a coordinate-free manner這些要素間的與

2、坐標(biāo)無關(guān)的數(shù)學(xué)運(yùn)算Define basic primitives基本的幾何體Line segments線段Polygons多邊形Outline4.1.1 Scalars標(biāo)量, Vectors向量, Points點4.1.2 Coordinate-free geometry這些要素間的與坐標(biāo)無關(guān)的幾何4.1.3The mathematic view: Vector and Affine Spaces向量空間與仿射空間4.1.6 Lines直線4.1.7 Affine Sums仿射加法4.1.8 Convexity凸性質(zhì)4.1.10 Planes平面4.2 Three-Dimensional pr

3、imitives三維圖元4.1.1 Scalars標(biāo)量, Vectors向量, Points點(Basic Elements基本幾何要素-1)Geometry is the study of the relationships among objects in an n-dimensional space幾何一門研究n維空間中對象之間的關(guān)系的學(xué)科In computer graphics, we are interested in objects that exist in three dimensions在計算機(jī)圖形學(xué)中，我們對三維空間中的對象感興趣Want a minimum set of

4、primitives from which we can build more sophisticated objects 通過基本幾何形狀的最小集合，根據(jù)這個集合可以建立起更復(fù)雜的對象4.1.1 Scalars標(biāo)量, Vectors向量, Points點(Basic Elements基本幾何要素-2)We will need three basic elements to describe the basic objects and the relationships among them需要三個幾何要素來描述這些基本幾何對象以及對象之間的關(guān)系Scalars標(biāo)量Vectors向量Points

5、點4.1.1 Scalars標(biāo)量, Vectors向量, Points點(Scalars標(biāo)量-1)Need three basic elements in geometry在幾何中需要三個基本元素Scalars, Vectors, Points標(biāo)量、向量、點Scalars can be defined as members of sets which can be combined by two operations (addition and multiplication) obeying some fundamental axioms (associativity, commutativi

6、ty, inverses)標(biāo)量可以定義為集合中的成員，集合中具有兩種運(yùn)算(加法和乘法)，運(yùn)算遵從一些基本的公理(結(jié)合律、交換律、逆) 4.1.1 Scalars標(biāo)量, Vectors向量, Points點(Scalars標(biāo)量-2)Examples include the real and complex number systems under the ordinary rules with which we are familiar例：實數(shù)或復(fù)數(shù)全體，通常的加法與乘法Scalars alone have no geometric properties標(biāo)量自身沒有幾何屬性4.1.1 Scala

7、rs標(biāo)量, Vectors向量, Points點(Vectors向量)Physical definition: a vector is a quantity with two attributes物理定義：向量是具有如下兩條性質(zhì)的量Direction方向Magnitude長度Examples include例Force力Velocity速度Directed line segments有向線段Most important example for graphics這也是圖形學(xué)中最重要的例子Can map to other types可以對應(yīng)到其它類型上v為什么需要向量？ -1 為什么需要向量？-

8、24.1.1 Scalars標(biāo)量, Vectors向量, Points點(Vector Operations向量運(yùn)算-1)Every vector has an inverse每個向量都有逆Same magnitude but points in opposite direction同樣長度但是指向相反的方向Every vector can be multiplied by a scalar每個向量都可以與標(biāo)量相乘v-vvvuw4.1.1 Scalars標(biāo)量, Vectors向量, Points點(Vector Operations向量運(yùn)算-2)There is a zero vector有一

9、個零向量Zero magnitude, undefined orientation零長度，方向不定The sum of any two vectors is a vector兩個向量的和為向量Use head-to-tail axiom三角形法則v-vvvuw4.1.2 Coordinate-Free Geometry與坐標(biāo)無關(guān)的幾何-1When we learned simple geometry, most of us started with a Cartesian approach初等幾何的學(xué)習(xí)中，主要應(yīng)用的是直角坐標(biāo)系Points were at locations in space

10、 p=(x,y,z)點在空間中的位置是p = (x,y,z)We derived results by algebraic manipulations involving these coordinates通過對這些坐標(biāo)進(jìn)行代數(shù)操作導(dǎo)出結(jié)果4.1.2 Coordinate-Free Geometry與坐標(biāo)無關(guān)的幾何-2This approach was nonphysical這種方法不是基于物理的Physically, points exist regardless of the location of an arbitrary coordinate system從物理的角度來講，點的存在性是

11、與坐標(biāo)系的具體位置無關(guān)的Most geometric results are independent of the coordinate system絕大多數(shù)幾何結(jié)果是不依賴于坐標(biāo)系的Example Euclidean geometry: two triangles are identical if two corresponding sides and the angle between them are identical歐氏幾何：兩個三角形全等是指它們有兩個對應(yīng)邊和夾角相等4.1.3 The mathematic view: vector and Affine Spaces(Affine

12、 Spaces仿射空間)在向量空間中，沒有位置和距離的概念仿射空間引入點，則有了位置的概念歐式空間（Euclid space）引入了距離，則有了距離的概念4.1.3 The mathematic view: vector and Affine Spaces(Affine Spaces仿射空間)Point + a vector space點加上向量構(gòu)造的空間Operations運(yùn)算：Vector-vector addition向量與向量的加法 向量Scalar-vector multiplication標(biāo)量與向量的乘法 向量Point-vector addition點與向量的加法 點Scalar

13、-scalar operations標(biāo)量與標(biāo)量的運(yùn)算標(biāo)量上述運(yùn)算均是與坐標(biāo)無關(guān)的For any point define對于任意點，定義1 P = P0 P = 0 (zero vector) (零向量)4.1.1 Scalars標(biāo)量, Vectors向量, Points點(Points點)Location in space空間中的位置用大寫字母表示Operations allowed between points and vectors點與向量之間可進(jìn)行的運(yùn)算Point-point subtraction yields a vector點與點相減得到一個向量Equivalent to poi

14、nt-vector addition等價地，點與向量相加得到新點P=v+Qv=P-Q4.1.3 The mathematic view: vector and Affine Spaces(Linear Vector Spaces線性向量空間)Mathematical system for manipulating vectors處理向量的數(shù)學(xué)系統(tǒng)Operations運(yùn)算Scalar-vector multiplication標(biāo)量乘法: u=vVector-vector addition向量加法: w=u+vExpressions such as v=u+2w-3r Make sense in

15、a vector space在向量空間中，表達(dá)式v = u + 2w 3r有意義4.1.3 The mathematic view: vector and Affine Spaces(Vectors Lack Position向量沒有位置)These vectors are identical向量是相等的Same length and magnitude具有相同的方向與長度Vectors spaces insufficient for geometry對幾何而言只有向量空間是不夠的Need points還需要點4.1.6 Lines直線Consider all points of the fo

16、rm考慮具有下述形式的所有點P(a)=P0 + a dSet of all points that pass through P0 in the direction of the vector d 即所有過P0點，與P0連線平行于向量d的點仿射空間中的直線4.1.6 Lines直線(Parametric Form參數(shù)形式)This form is known as the parametric form of the line上述定義直線的形式稱為參數(shù)形式More robust and general than other forms比其它形式更一般和穩(wěn)定Extends to curves a

17、nd surfaces可以推廣到曲線和曲面Two-dimensional forms二維形式Explicit顯式: y = mx +hImplicit隱式: ax + by +c =0Parametric參數(shù)形式: x(a) = ax0 + (1-a)x1 y(a) = ay0 + (1-a)y14.1.7 Affine Sums仿射加法(Rays and Line Segments射線與線段)If a = 0, then P(a) is the ray leaving P0 in the direction d如果限定 0, 那么P()就是從P0出發(fā)，方向為d 的射線 If we use t

18、wo points to define v, then如果采用兩點定義向量d, 那么 P( a) = Q + a (R-Q) = Q+av = aR + (1-a)QFor 0=a=0, we have the convex hull of P1,P2,.Pn 進(jìn)一步，如果i0，那么得到P1, P2 , Pn的凸包(convex hull)4.1.8 Convexity(Convex Hull凸包)Smallest convex object containing P1,P2,.Pn 最小的包含P1, P2 , Pn的凸體Formed by “shrink wrapping” points 可

19、以用“收縮包裝”的方式得到4.1.8 Convexity (Convexity凸體)An object is convex iff for any two points in the object all points on the line segment between these points are also in the object一個對象為凸的當(dāng)且僅當(dāng)在對象中任何兩點的連接線段也在該對象內(nèi)PQQPconvex凸not convex凸4.1.8 Convexity(Curves and Surfaces曲線與曲面)Curves are one parameter entities

20、of the form P(a) where the function is nonlinear曲線是形式為P()的單參數(shù)定義的幾何體，其中的函數(shù)為非線性Surfaces are formed from two-parameter functions P(a, b)曲面是由形式為P(, )的兩個參數(shù)定義的幾何體體Linear functions give planes and polygons線性函數(shù)對應(yīng)于 平面 和 多邊形P(a)P(a, b)4.1.9 Dot products and cross products點積和叉積(內(nèi)積和外積)4.1.10 Planes平面A plane can

21、 be defined by a point and two vectors or by three points平面是由一個點與兩個向量或者三個點確定的P(a,b) = R+ au + bvP(a,b) = R + a(Q-R) + b(P-Q)uvRPRQ4.1.10 Planes平面(Triangles三角形)convex sum of P and QP與Q的凸組合convex sum of S(a) and RS()與R的凸組合for 0=a,b=1, we get all points in triangle當(dāng)0, 1時定義在三角形內(nèi)的點4.1.10 Planes平面(Normals法向量)Every plane has a vector n normal (perpendicular, orthogonal) to it