A good introductory book to geodesics, curve length, curvature. I'm looking for a book about geodesics, curve length, curvature. Preferably something around undergraduate level. 
I'm doing a project where we have to show that a simulation we run, plots a geodesic on a sphere. So something that can give me the basics would be a helpful start.
Thanks.
 A: There are tons of choice. Many people like Pressley's Elementary Differential Geometry (used in my college too), but i'm not really a fan. 
There are several books for undergraduate level which i found very interesting and "complete". Some of my favorites are: 
M. Abate & F. Tovena - "Curves and Surfaces" is a very well written text which also includes many advanced topic of differential geometry (on manifolds) where normally it's not introduced on undergrad level.  
Christian Bar - "Elementary Differential Geometry " which included many good illustration and colored picture in the middle, 
Antonio Ros & Sebastian Montiel - "Curves and Surfaces". 
Abbena, Salamon and Gray - "Modern Differential Geometry of Curves and Surfaces with Mathematica"  if you are Mathematica's fan. 
Also, you can try the classics like Do Carmo's Differential Geometry of Curves and Surfaces (cheaper in dover edition now) or O'neill's Elementary Differential Geometry. There is also an online lecture notes and youtube video course (very good) by Prof. Shifrin here and here. Good Luck !
A: Several of the texts that have been recommended are intended for graduate work. I'll add my own text to the list. For one thing, it's free and immediately downloadable. :) Check sections 1.1, 1.2, and 2.4 (although you'll probably need a bit of notation from earlier sections). By the way, geodesics on the sphere are easy — they're just the great circles. :)
A: Try An introduction to Differential Geometry by Willmore. This book may also be helpful Introduction to Manifolds  by  Taiwanese-American mathematician Loring W. Tu .
A: Loring Tu has made a couple books on these topics and a LOT more. The two books are called 1) An Introduction to Manifolds and 2) Differential Geometry.
These are the gold standards when it comes to detail and intuition in my opinion. Since they are new ( 2 was released this year!), they are a priori one of the most up to date modern texts you can read. They should be read in order (if you know some manifold theory, then you can skip to the second book).
Everything is abstractized in order from the standard 2 and 3 dimensional case to $n$-dimensions. Sometimes he discusses some general notions first, then specializes, but every general notion he defines is usually incredibly important not only as a tool for the subject itself, but a tool in general mathematics (a lot of applications in algebraic geometry, algebraic number theory, etc).
