L8 – Differential Geometry of Surfaces

• We will discuss intrinsic properties of surfaces independent its parameterization • Topics to be covered including: – Tangent plane and surface normal – First fundamental form I (metric) – Second fundamental form II (curvature) – Principal curvatures – Gaussian and mean curvatures – Euler’s theorem

1 Tangent Plane and Surface Normal

• Consider a curve u=u(t) v=v(t) in the parametric domain of a parametric surface r=r(u, v), then r=r(t)=r(u(t), v(t)) is a parametric curve lying on the surface r. z r(u, v) v Using the chain rule on r(u(t), v(t)) r(t)=r(u(t), v(t))

u=u(t) 풓ሶ 푡 = 풓푢푢ሶ + 풓푣푣ሶ v=v(t) o y u x • The tangent plane at point p can be considered as a union of the tangent vectors of the form in above equation for all r(t) (i.e., many curves) passing through p. z N

rv 풓푢 × 풓푣 ru 푵 = 풓푢 × 풓푣

o y

x 2 Tangent Plane and Surface Normal (cont.)

• The equation of the tangent plane at r(up, vp) is

(p - r(up, vp)) ∙ N(up, vp) =0 • Definition: A regular (ordinary) point p on a parametric surface is

defined as a point where ru x rv ≠ 0. A point with ru x rv = 0 is called a singular point.

• Implicit surface f(x,y,z)=0, the unit normal vector is given by 훻푓 푵 = 훻푓

3 First Fundamental Form I (Metric)

• If we define the parametric curve by a curve u=u(t) v=v(t) on a parametric surface r=r(u,v) 푑풓 푑푢 푑푣 푑푠 = 푑푡 = 풓 + 풓 푑푡 푑푡 푢 푑푡 푣 푑푡

= (풓푢푢ሶ + 풓푣푣ሶ) ∙ (풓푢푢ሶ + 풓푣푣ሶ)푑푡 푑푢 푑푣 = 퐸푑푢2 + 2퐹푑푢푑푣 + 퐺푑푣2 (∵ 푢ሶ = , 푣ሶ = ) 푑푡 푑푡 where

퐸 = 풓푢 ∙ 풓푢, 퐹 = 풓푢 ∙ 풓푣, 퐺 = 풓푣 ∙ 풓푣 • E, F, G are call the first fundamental form coefficients and play important role in many intrinsic properties of a surface

4 First Fundamental Form I (cont.)

• Definition: The first fundamental form is defined as 퐼 = (푑푠)2= 푑풓 ∙ 푑풓 = 퐸푑푢2 + 2퐹푑푢푑푣 + 퐺푑푣2 which can also be rewritten as 푎 × 푏 ∙ 푐 × 푑 1 퐸퐺 − 퐹2 =(a ∙ c)(b ∙ d)-(a ∙ d)(b ∙ c) 퐼 = (퐸푑푢 + 퐹푑푣)2+ (푑푣)2 퐸 퐸 • We can then have 퐸 = 풓푢 ∙ 풓푢 > 0 2 (풓푢 × 풓푣) = 풓푢 × 풓푣 ∙ 풓푢 × 풓푣 2 2 = 풓푢 ∙ 풓푢 풓푣 ∙ 풓푣 − 풓푢 ∙ 풓푣 = 퐸퐺 − 퐹 > 0 • Conclusion: we know that I is positive definite, provided that the surface is regular (since 퐼 ≥ 0 and I = 0 if and only if du=0 and dv=0 at the same time) 5 Example I: Let’s compute the arc length of a curve 푢 = 푡, 푣 = 푡 for 0 ≤ 푡 ≤ 1 on a hyperbolic paraboloid r(u, v)=(u, v, uv)T where 0 ≤ 푢, 푣 ≤ 1. 푇 푇 Solution: 풓푢 = (1,0, 푣) , 풓푣 = (0,1, 푢) 2 2 퐸 = 풓푢 ∙ 풓푢 = 1 + 푣 , 퐺 = 풓푣 ∙ 풓푣 = 1 + 푢 , 퐹 = 풓푢 ∙ 풓푣 = 푢푣. and along the curve the first fundamental form coefficients are 퐸 = 1 + 푡2, 퐺 = 1 + 푡2, 퐹 = 푡2. Thus

1 푑푠 = 퐸푢ሶ 2 + 2퐹푢ሶ 푣ሶ + 퐺푣ሶ 2푑푡 = 2 + 4푡2푑푡 = 2 푡2 + 푑푡 2 The arc length 1 1 1 1 1 1 ( + 푠 = ׬ 2 푡2 + 푑푡 = 푡 푡2 + + log(푡 + 푡2 0 2 2 2 2 0 3 1 = + log( 2 + 3) 2 2 6 First Fundamental Form I (Angle)

• The angle between two curves on a parametric surface

풓1 = 풓(푢1 푡 , 푣1 푡 ), 풓2 = 풓(푢2 푡 , 푣2 푡 ) can be evaluated by tacking the inner product of their tangent vectors. 푑풓 ∙ d풓 cos 휔 = 1 2 푑풓1 푑풓2 퐸푑푢 푑푢 + 퐹 푑푢 푑푣 + 푑푣 푑푢 + 퐺푑푣 푑푣 = 1 2 1 2 1 2 1 2 2 2 2 2 퐸푑푢1 + 2퐹푑푢1푑푣1 + 퐺푑푣1 퐸푑푢2 + 2퐹푑푢2푑푣2 + 퐺푑푣2 • As a result, the orthogonality condition for 풓ሶ 1 and 풓ሶ 2 is 퐸푑푢1푑푢2 + 퐹 푑푢1푑푣2 + 푑푣1푑푢2 + 퐺푑푣1푑푣2 = 0 • In particular, the angle of two iso-parametric curves is 풓 ∙ 풓 풓 ∙ 풓 퐹 cos 휔 = 푢 푣 = 푢 푣 = 풓푢 풓푣 풓푢 ∙ 풓푢 풓푣 ∙ 풓푣 퐸퐺 Thus, the iso-parametric curves are orthogonal if 퐹 ≡ 0. 7 First Fundamental Form I (Area)

• The area of a small parallelogram with 4 vertices 풓(푢, 푣), 풓(푢 + 훿푢, 푣), 풓(푢, 푣 + 훿푣), 풓(푢 + 훿푢, 푣 + 훿푣) 풓(푢, 푣 + 훿푣) 풓(푢 + 훿푢, 푣 + 훿푣) δ퐴 풓(푢, 푣) 풓(푢 + 훿푢, 푣) is approximated by 2 δ퐴 = 풓푢훿푢 × 풓푣훿푣 = 퐸퐺 − 퐹 훿푢훿푣 2 2 (∵ 풓푢 × 풓푣 =(풓푢 ∙ 풓푢)(풓푣 ∙ 풓푣) − (풓푢 ∙ 풓푣) ) Or in different form as 푑퐴 = 퐸퐺 − 퐹2푑푢푑푣 8 Example II: Let’s compute the area of a region on the hyperbolic paraboloid. The region is bounded by positive u and v axes and a quarter circle 푢2 + 푣2 = 1 as shown. Solution: Again 푇 푇 풓푢 = (1,0, 푣) , 풓푣 = (0,1, 푢) 2 2 퐸 = 풓푢 ∙ 풓푢 = 1 + 푣 , 퐺 = 풓푣 ∙ 풓푣 = 1 + 푢 , 퐹 = 풓푢 ∙ 풓푣 = 푢푣

퐸퐺 − 퐹2 = 1 + 푣2 1 + 푢2 − 푢2푣2 = 1 + 푢2 + 푣2

∴ 퐴 = න 1 + 푢2 + 푣2푑푢푑푣 퐷 Letting 푢 = 푟 cos 휃 and 푣 = 푟 sin 휃 휋 2 1 휋 퐴 = න න 푟 1 + 푟2 푑푟푑휃 = ( 8 − 1) 0 0 6

9 Second Fundamental Form II (Curvature)

• To quantify the curvature of a surface S, we consider a curve C on S which pass through the point p. N • The unit tangent vector t and t n the unit normal vector n of the kn k S C k curve C are related by g

k = dt/ds = kn = kn + kg where kn is the normal curvature vector and kg is the geodesic curvature vector. kg : the component of k in the direction perpendicular to t in the surface tangent plane kn : the component of k in the surface normal direction kn = knN, where kn - the normal curvature of surface at p in t direction. 10 Second Fundamental Form II (cont.)

By differentiating 푵 ∙ 풕 = 0 along the curve with respect to s, we have 푑풕 푑푵 ∙ 푵 + 풕 ∙ = 0 푑푠 푑푠 Thus 푑풕 푑푵 푑풓 푑푵 ∵ 푑풓 = 풓푢푑푢 + 풓푣푑푣 푘푛 = ∙ 푵 = −풕 ∙ = − ∙ 푑푵 = 푵푢푑푢 + 푵푣푑푣 푑푠 푑푠 푑푠 푑푠 ∴ 푑풓 ∙ 푑푵 = 풓 ∙ 푵 푑푢2 퐿푑푢2 + 2푀푑푢푑푣 + 푁푑푣2 푢 푢 + 풓푢 ∙ 푵푣 + 풓푣 ∙ 푵푢 푑푢푑푣 = 2 2 2 퐸푑푢 + 2퐹푑푢푑푣 + 퐺푑푣 + 풓푣 ∙ 푵푣 푑푣 with

퐿 = −풓푢 ∙ 푵푢, 푁 = −풓푣 ∙ 푵푣, 1 푀 = − 풓 ∙ 푵 + 풓 ∙ 푵 = −풓 ∙ 푵 = −풓 ∙ 푵 2 푢 푣 푣 푢 푢 푣 푣 푢

11 Second Fundamental Form II (cont.)

• Since ru and rv are perpendicular to N, we have 풓푢 ∙ 푵 = 0 ⟹ 풓푢푢 ∙ 푵 + 풓푢 ∙ 푵푢 = 0 (for L) 풓푣 ∙ 푵 = 0 ⟹ 풓푣 ∙ 푵푢 + 풓푢푣 ∙ 푵 = 0 (for M) 풓푣 ∙ 푵 = 0 ⟹ 풓푣푣 ∙ 푵 + 풓푣 ∙ 푵푣 = 0 (for N) hence: 퐿 = 풓푢푢 ∙ 푵, 푀 = 풓푢푣 ∙ 푵, 푁 = 풓푣푣 ∙ 푵 • We define the second fundamental form II as 퐼퐼 = 퐿푑푢2 + 2푀푑푢푑푣 + 푁푑푣2 and L, M, N are called second fundamental form coefficients 퐼퐼 퐿 + 2푀휆 + 푁휆2 푘 = = 푛 퐼 퐸 + 2퐹휆 + 푁퐺2 푑푣 where 휆 = is the direction of the tangent line to C at p. 푑푢 12 • Theorem: All curves lying on a surface S passing through a given point 풑 ∈ 푆 with the same tangent line have the same normal curvature at this point. • Note that, sometimes the positive normal curvature is defined in the opposite direction.

• We now start to study the local surface property by II. • Suppose two neighboring points P and Q on S 푷 = 풓(푢, 푣) and 푸 = 풓(푢 + 푑푢, 푣 + 푑푣) By Taylor expansion, we have 풓 푢 + 푑푢, 푣 + 푑푣 = 풓 푢, 푣 + 풓푢푑푢 + 풓푣푑푣 1 + 풓 푑푢2 + 2풓 푑푢푑푣 + 풓 푑푣2 + ⋯ 2 푢푢 푢푣 푣푣 13 • Projecting the vector: 푷푸 = 풓 푢 + 푑푢, 푣 + 푑푣 − 풓 푢, 푣 1 = 풓 푑푢 + 풓 푑푣 + 풓 푑푢2 + 2풓 푑푢푑푣 + 풓 푑푣2 + ⋯ 푢 푣 2 푢푢 푢푣 푣푣 onto N by using the second fundamental form, we have 1 1 푷푸 ∙ 푵 = (풓 푑푢 + 풓 푑푣) ∙ 푵 + 퐼퐼 = 퐼퐼 푢 푣 2 2 (∵ 풓푢 ∙ 푵 = 풓푣 ∙ 푵 = 0) where the higher order terms are neglected. • In conclusion, the local shape variation: d = II / 2 When d = 0, it become 퐿푑푢2 + 2푀푑푢푑푣 + 푁푑푣2 = 0 which can be considered as a quadratic equation in terms of du, dv. −푀± 푀2−퐿푁 푑푢 = 푑푣 (Assuming 퐿 ≠ 0) 퐿 14 Local Shape Analysis by Second Fundamental Form II • 푀2 − 퐿푁 < 0, this is no real root: No intersection between r(u,v) and Tp except the point p. An elliptic point

• 푀2 − 퐿푁 = 0 and 퐿2 + 푀2 + 푁2 ≠ 0, there are double roots: The surface intersects Tp with one line 푀 푑푢 = − 푑푣 퐿 A parabolic point • 푀2 − 퐿푁 > 0, there are two roots: The surface intersects its tangent lane with two lines −푀± 푀2−퐿푁 푑푢 = 푑푣 퐿

A hyperbolic point 15 Local Shape Analysis by Second Fundamental Form II (cont.) • L = M = N = 0, the surface and the tangent plane have a contact of higher order than in the above cases at the point p. A flat or planar point • If L = 0 and 푁 ≠ 0, we can solve for dv instead of du. • If L = N = 0 and 푀 ≠ 0, we have 2푀푑푢푑푣 = 0, thus the intersection lines will be the two iso-parametric lines: u = const1 v = const2

16 Principal Curvatures

• From the second fundamental form, we already know 퐼퐼 퐿+2푀휆+푁휆2 푘 = = 푛 퐼 퐸+2퐹휆+푁퐺2 푑푣 The normal curvature at p depends on the direction of 휆 = . 푑푢

• The extreme value of 푘푛 can be obtained by solving: 푑푘푛/푑휆 = 0. 퐸 + 2퐹휆 + 퐺휆2 푁휆 + 푀 − 퐿 + 2푀휆 + 푁휆2 퐺휆 + 퐹 = 0 퐿+2푀휆+푁휆2 푁휆+푀 푘푒 = = 푛 퐸+2퐹휆+퐺휆2 퐺휆+퐹 • Furthermore, since 퐸 + 2퐹휆 + 퐺휆2 = 퐸 + 퐹휆 + 휆 퐹 + 퐺휆 and 퐿 + 2푀휆 + 푁휆2 = 퐿 + 푀휆 + 휆 푀 + 푁휆 , we have 퐸 + 퐹휆 + 휆 퐹 + 퐺휆 푁휆 + 푀 = 퐿 + 푀휆 + 휆 푀 + 푁휆 퐺휆 + 퐹 퐸 + 퐹휆 푁휆 + 푀 = 퐿 + 푀휆 퐺휆 + 퐹

17 • Therefore, we have 퐿 + 2푀휆 + 푁휆2 푀 + 푁휆 퐿 + 푀휆 푘푒 = = = 푛 퐸 + 2퐹휆 + 퐺휆2 퐹 + 퐺휆 퐸 + 퐹휆 푑푣 • Substituting 휆 = , we have 푑푢 푒 푒 퐿 − 푘푛퐸 푑푢 + 푀 − 푘푛퐹 푑푣 = 0 푒 푒 푀 − 푘푛퐹 푑푢 + 푁 − 푘푛퐺 푑푣 = 0 This is a homogeneous linear system of equations for du, dv, which will have a nontrivial solution if and only if 푒 푒 퐿 − 푘푛퐸 푀 − 푘푛퐹 푒 푒 = 0 푀 − 푘푛퐹 푁 − 푘푛퐺 2 푒2 푒 2 퐸퐺 − 퐹 푘푛 − 퐸푁 + 퐺퐿 − 2퐹푀 푘푛 + 퐿푁 − 푀 = 0 푒 • The discriminant of D of this quadratic equation in 푘푛 can be 퐸퐺 − 퐹2 2퐹 2 퐷 = 4 퐸푀 − 퐹퐿 2 + 퐸푁 − 퐺퐿 − (퐸푀 − 퐹퐿) 퐸2 퐸 Since 퐸퐺 − 퐹2 ≥ 0, we have 퐷 ≥ 0. The quadratic equation has real roots.

18 • If we set 퐿푁−푀2 퐸푁+퐺퐿−2퐹푀 퐾 = and 퐻 = 퐸퐺−퐹2 2(퐸퐺−퐹2) The quadratic equation is simplified to 푒2 푒 푘푛 − 2퐻푘푛 + 퐾 = 0 which has the following solutions 2 푘max = 퐻 + 퐻 − 퐾 2 푘min = 퐻 − 퐻 − 퐾

• kmax and kmin are defined as the maximum and minimum principal curvature.

• The directions for which give kmax and kmin are called principal directions, the corresponding directions in uv-plane 푀−푘 퐹 퐿−푘 퐸 휆 = − 푛 or − 푛 푁−푘푛퐺 푀−푘푛퐹 by replacing kn with either kmax or kmin. 19 2 • When H = K, kn is a double root with value equal to H and the corresponding point of the surface is an umbilical point. – At an umbilical point, a surface is locally a part of sphere with radius of curvature: 1 / H – In the special case – both K and H vanish, the point is a flat or planar point • A curve on surface where tangent at each point is in a principal direction at that point is called a line of curvature – At each (non-umbilical) point, there are two principal directions that are orthogonal – the lines of curvatures form an orthogonal net of lines. – We can try to use this orthogonal net of lines for parameterization 휋 – Let cos 휔 = cos = 0 (page 7), we have F = 0 2 – We can then from the equation of 휆 have

퐿 − 푘푛퐸 푑푢 + 푀푑푣 = 0 푀푑푢 + 푁 − 푘푛퐺 푑푣 = 0 • The necessary condition for the parametric lines

to be line of curvature is: F = M = 0. Line of curvatures Gaussian and Mean Curvatures

• From above equation on page 19, we can easily have 퐾 = 푘max푘min 퐻 = 푘min + 푘max /2 • The sign of the Gaussian curvature coincides with the sign of 퐿푁 − 퐿푁−푀2 푀2, since 퐾 = and 퐸퐺 − 퐹2 > 0. Then, we have 퐸퐺−퐹2 1. 퐾 > 0, the surface is elliptic at the point; 2. 퐾 < 0, the surface is hyperbolic at the point; 3. 퐾 = 0 and 퐻 ≠ 0, the surface is parabolic at the point; 4. 퐾 = 0 and 퐻 = 0, the surface is flat (or planar) at the point. • This is good for segmenting an given surface to different regions (reverse engineering of CAD models). http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.83.3315&rep=rep1&type=pdf 21 Euler’s Theorem

• The normal curvature of a surface in an arbitrary direction (in the tangent plane) at point p can be expressed in terms of principal

curvatures k1 and k2 at point p and the angle between the arbitrary direction and the principal direction corresponding to k1 as: 2 2 푘푛 휙 = 푘1 cos 휙 + 푘2 sin 휙

22