Making future goals clear in statement of purpose when changing fields I need your advice, dear visitors. Currently I am finishing my BS in Applied Mathematics and Computer Science, and trying to apply for a PhD program in Computer Science to universities in the USA.

But I have a real problem. 

The topics on which I worked with my adviser are purely mathematical (differential geometry). I began to study this branch of math 2 years ago. But 1 year ago, due to a suggestion of my friend, I started reading articles on Theoretical Computer Science (namely algorithms on planar graphs and in Linear Algebra but nothing related to numerical methods) and became carried away by that. For some reasons, I couldn't have abandoned the work on my topics. One of them is that the results in differential geometry we eventually obtained are excellent (together with my advisor we have two articles in good mathematical journals). Since this is the last year of my education at my University, I haven't changed the field for my diploma (it would have had no sense and it would have been impossible in some way, but let me not tell about it). 

Differential Geometry is very interesting and I know much about it but I feel that algorithms is the field to which my future research will be dedicated. I am not interested in any other field of CS. By the way, my scientific advisor knows about my future plans (reported to him half a year ago). He also heartily agreed to write a strong letter of recommendation for me.

So what have I got? Good mathematical skills: a wide spectrum of knowledge on many fields including algebra (linear and abstract), advanced analysis, differential geometry (as mentioned before), functional analysis, combinatorics, algorithms, graph theory... I also have medals and prizes earned on mathematical olympiads (IMO, IMC). So math and TCS background is not a problem for me. After all, Theoretical Computer Science is the same math (we also state and prove theorems like in math)!

Why a PHD? Because I absolutely know that one-two years spent on getting a MS finished will be just a waste of my time.

So I feel difficulty when composing a Statement of Purpose. I just don't know in which way I should put the description of my personal experience forward in such a way so it would be clear and laconic. I don't want the reader to be confused about my goals. I don't want my statement of purpose to be a fruit-salad. I want the reader to understand that a PHD in CS is what I need since I am completely determined about my intentions. 

A lot of sincere thanks to whoever will answer my question. Maybe, one can give me a link to a sample of SOP similar to my case.
I think you've essentially written an outline of your SoP in the question. Namely, 

"I'm interested in theoretical computer science and mathematics. I've worked in differential geometry, and here are my results, but I'm now fascinated by algorithms research, and I have extensive math background". 

Now just expand on it.