The term curvature simply refers to the amount of bending of a

curve or a surface. In simple terms, curvature at a point indicates

how quickly a curve or a surface bends away from its local tangent

line or tangent plane, respectively. While tangent planes are useful

for studying first-order behavior of surfaces, curvatures deal with

second-order behavior. However, a rigorous study of curvatures

can be quite involved, and in general, there are different notions

of curvatures depending on the context. The interested reader is

referred to the comprehensive survey of curvature in the classical

differential geometry text by Do Carmo [dC76]. In this chapter,

we will explain the basic ideas, develop some intuitions, and study

the necessary formulations that allow us to better understand the

notion of curvature and help us to compute various types of cur-