2018/3/11 Explicit/parametric and implicit definition 图上活大学 Expanded into three dimensions curves in two dimensions 国上清大学 ty Both implicit and explicit forms Explicit form can be extended into three function dimensional shapes y=f(x) -A three dimensional curve =f(x.y) However,a multi-valued curve wou beo euing -A spherical surface located at (a.be) Implicit form (x-a2+0y-b2+2-c)2=r2 Implicit forms are more powerful in terms of the types of curves/surfaces it can represent Explicit forms are easy to implement on computers oorn enin Song Parametric form of a curve 国上活大坐 Parametric form of a surface 圈上活大整 A two/three dimensional curve can also be defined using parametric form metricace cnbe [x=f(u) f(uv)=(/(uv).f(uv)./(uv)) y=∫(u) A sphere of radius R located at (a,b,c) =f(u) can be given by [x=a+Rcosycos0 y=b+Rcosysin =c+Rsiny 「z Basic Airplane shape design 圆上清文大等 Airfoil Parameterization(1) 园上海发大坐 ·Joukowski airfoils Defined by Joukowski transformation, Buttock-planes method w(e=:+1/: Radiusography trouintdescribesa circle 出 Method of Conic curves ·Airfoil families NACA 4-series ,NURBS curves NACA 6-series -etc. For example r--.117,g-.048 Radins-1124 Shanghal Jao Tong Uniersty-Dr.Wenbin Sor rslty-Dr.V(enbin Song 112018/3/11 11 © Shanghai Jiao Tong University – Dr. Wenbin Song School of Aeronautics and Astronautics Explicit/parametric and implicit definition of curves in two dimensions • A two-dimensional curve can be defined as a single valued function • However, a multi-valued curve would be easier to define using • Implicit forms are more powerful in terms of the types of curves/surfaces it can represent • Explicit forms are easy to implement on computers y f x  ( ) x y x y 2 2 2 ( ) x b  a   ( ) y r  Explicit form Implicit form © Shanghai Jiao Tong University – Dr. Wenbin Song School of Aeronautics and Astronautics Expanded into three dimensions • Both implicit and explicit forms can be extended into three dimensional shapes – A three dimensional curve – A spherical surface located at (𝑎, 𝑏, 𝑐) z f x, y  ( ) 2 2 2 2 ( ) + ) x a      ( ) (z y b c r x y z © Shanghai Jiao Tong University – Dr. Wenbin Song School of Aeronautics and Astronautics Parametric form of a curve • A two/three dimensional curve can also be defined using parametric form (u) (u) (u) x y z x f y f z f         x y x y z © Shanghai Jiao Tong University – Dr. Wenbin Song School of Aeronautics and Astronautics Parametric form of a surface • A generic parametric surface can be defined as • A sphere of radius 𝑅 located at 𝑎, 𝑏, 𝑐 can be given by x y z (u,v) ( (u,v), (u,v), (u,v)) x y z f f f f  cos cos cos si s n in x a R y b R z c R                 © Shanghai Jiao Tong University – Dr. Wenbin Song School of Aeronautics and Astronautics Basic Airplane shape design • Buttock-planes method • Radiusography • Method of Conic curves • Spline, NURBS curves methods © Shanghai Jiao Tong University – Dr. Wenbin Song School of Aeronautics and Astronautics Airfoil Parameterization (1) • Joukowski airfoils – Defined by Joukowski transformation, – where is a complex number, and if z describes a circle that passes through point 1 in the complex plane • Airfoil families – NACA 4-series – NACA 6-series – etc. z  x  iy wz  z 1/ z For example