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Award-Winning Affine geometry Tutors

Certified Tutor
10+ years
Aaron
I'm not tutoring or buried in my textbooks, you will either find me rock climbing at the Triangle Rock Club, playing Ultimate Frisbee, working on my car, or enjoying the great outdoors (beaches, mountains, forests--you name it, I love it). On rainy weekends I enjoy tinkering with computers and old e...
The University of Texas at Dallas
Bachelors, Mechanical Engineering
Duke University
Current Grad Student, Mechanical Engineering

Certified Tutor
6+ years
Mimi
I am an interdisciplinary educator with an Ed.M. from the Harvard Graduate School of Education and a B.A. from Dartmouth College. My background is primarily in integrated arts learning and museum education and I specialize in visual arts, history and art history, and object-based learning. In all su...
Harvard University
Masters in Education, Education
Dartmouth College
B.A.
Certified Tutor
10+ years
Nina
I am a recent graduate from a masters program in biostatistics at Columbia University. I received my Bachelor of Arts in biological sciences, with a focus in neurobiology at Northwestern University. In August, I will be starting a doctoral program in biostatistics at NYU. I was a teaching assistant ...
Columbia University
Masters in biostatistics
Northwestern University
Bachelor of Arts in biological sciences (focus in neurobiology)
Columbia University in the City of New York
Current Grad Student, Biostatistics
Certified Tutor
I am a graduate of Wesleyan University, where I received my Bachelor of Arts in Sociology with High Honors. With eight years of experience working in education, I've tutored students in math, science, history, and English, as well as helped students prepare for standardized tests. I've guided adults...
Harvard University
PHD, Education
Wesleyan University
Bachelor in Arts, Sociology
Certified Tutor
Christopher
I am a rising sophomore at Harvard College and am about to declare as a Mechanical Engineering concentrator, working towards a Bachelor of Science degree. I've always enjoyed sharing my knowledge with my peers and those around me and have done so in both formal and informal settings. I've been a tut...
Harvard College
Bachelor of Science, Mechanical Engineering
Certified Tutor
Charles
I am a junior Mechanical Engineering major at Yale, and I hope to become a Naval Aviator after college. I am also a varsity sailor, and enjoy playing music with friends when I can get some free time. I have been tutoring my fellow students throughout my entire academic career, and I would best descr...
Yale University
Bachelor of Science, Mechanical Engineering
Certified Tutor
Liz
I am a graduate of Washington University in St Louis, where I received my Bachelor of Arts in History with minors in Humanities and Anthropology. Since graduation, I have worked as a tutor, teacher, and director of tutors at a charter public middle school in Boston. During this time I also received ...
Simmons College
Masters, Special Education: Mild to Moderate Disabilities 5-12
Washington University in St. Louis
Bachelor of Arts in History (minors in Humanities and Anthropology)
Certified Tutor
Michelle
I am proud to be a part of Varsity Tutors! I am originally from San Antonio, TX; I completed my undergraduate education at Rice University in Houston where I received a bachelor's degree in Biochemistry and Cell Biology. Currently, I am in my second year of medical school at Baylor College of Medici...
Baylor College of Medicine
Current Grad Student, M.D.
Rice University
Bachelor's in Biochemistry and Cell Biology
Certified Tutor
8+ years
Solange
I'm Solange - a recent graduate from Harvard where I studied Sociology & Women's Studies. I've been tutoring for eight years now, and have worked with a wide range of ages and in a wide range of subjects. Some of my specialties are college prep/test taking II worked in the admissions office on campu...
Harvard University
Bachelor in Arts (Sociology & Women's Studies)
Certified Tutor
9+ years
Justin
I am an aspiring applied mathematician, with particular interest in image processing and climate science. I graduated in May 2017 from Washington University in St. Louis with a bachelor's in physics and mathematics, and am beginning a PhD program in September 2017 at the University of Chicago in Com...
Washington University in St. Louis
Bachelor's in Physics and Mathematics
University of Chicago
Doctor of Philosophy, Computational Mathematics
Certified Tutor
9+ years
Henry
I'm eager to help you in your education. I'm a recent graduate of Harvard College looking to apply to law school. My senior thesis was written on John Dewey's ideas of education, which I deeply believe has incredible power to transform individuals and society.
Harvard College
Bachelor in Arts, History
Certified Tutor
Elena
I am a graduate of McGill University (BA First Class Honors) and the University of Edinburgh (MSc First Class Honors with Distinction) with over eight years of tutoring experience. I am currently a curriculum developer for a company which creates relatable and culturally-literate courses for middle ...
University of Edinburgh
Masters, Biblical Studies
Mcgill University
Bachelor in Arts, Religious Studies
Certified Tutor
Justin
I am a graduate of the University of Chicago where I received my Bachelor of Arts in Philosophy. Currently, I am in the master's program at the University of New Mexico where I am continuing my education in philosophy. Ultimately, I hope to go on to earn a PhD in Philosophy so that I can continue en...
University of Chicago
Bachelor of Arts in Philosophy
University of New Mexico-Main Campus
Current Grad Student, Philosophy
Certified Tutor
9+ years
Isabella
I am a graduate of MIT. I received my Bachelor of Science in Mathematics with minors in Management Science and Ancient and Medieval Studies. Since graduation, I have started my PhD at Georgia Tech in Operations Research. Throughout my career I have TA'd several math and computer science courses at t...
Massachusetts Institute of Technology
Bachelor of Science in Mathematics (minors in Management Science and Ancient and Medieval Studies)
Georgia Institute of Technology-Main Campus
Current Grad Student, Operations Research
Certified Tutor
6+ years
Ingrid
I am exploring my creativity by pursuing a double major in Asian Languages and Cultures with a focus in Korean, studying abroad in South Korea as a Benjamin A. Gilman Scholar, leading workshops that teach 3D printing and CAD for undergraduate students as the president of 3D4E, advocating for the fir...
Northwestern University
Bachelor of Science, Biomedical Engineering
Top 20 Math Subjects
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Henry
Calculus Tutor • +41 Subjects
I'm eager to help you in your education. I'm a recent graduate of Harvard College looking to apply to law school. My senior thesis was written on John Dewey's ideas of education, which I deeply believe has incredible power to transform individuals and society.
Elena
Calculus Tutor • +31 Subjects
I am a graduate of McGill University (BA First Class Honors) and the University of Edinburgh (MSc First Class Honors with Distinction) with over eight years of tutoring experience. I am currently a curriculum developer for a company which creates relatable and culturally-literate courses for middle and high-schools, and am particularly adept at communicating and explaining concepts in a quirky, engaging, and intelligent manner. I was named Scotland International Young Thinker of the Year 2014 for exactly that sort of work. Much of my tutoring background is in test-prep and essay coaching, which I enjoy because it allows the tutor and student to think strategically together, and work as a team to achieve concrete results. I have worked with students ranging in age from 6-32, and believe that, in an educational context, a few jokes never hurt anybody. I love reading and learning, and my educational approach is centered around making the material just as engaging to students as it is to me. I think J.K. Rowlings, the writer of Harry Potter, is just as brilliant as Stephen Hawking, and in my free time, I manage my (terrible) fantasy baseball team, write songs for my comedy band, and crack jokes about terrible science-fiction movies with my friends.
Justin
Calculus Tutor • +38 Subjects
I am a graduate of the University of Chicago where I received my Bachelor of Arts in Philosophy. Currently, I am in the master's program at the University of New Mexico where I am continuing my education in philosophy. Ultimately, I hope to go on to earn a PhD in Philosophy so that I can continue engaging in my passions for learning and teaching. While in school, I have spent countless hours coaching high school speech and debate both in person and working online with students across the country. My focus in coaching has been to emphasize philosophy and critical thought to prepare students to think through novel arguments on their own. I am passionate about teaching and tutoring because I love seeing students learn to be intellectually independent and think through problems on their own terms by developing their critical thinking skills. I have devoted my life to education because I am passionate about it, and I try to share some of my passion for learning with the students I work with. I tutor all sorts of Standardized Tests, and I particularly enjoy working on logic-based problems like analogies and math sections. When I am not tutoring or reading for school, I enjoy strategy games (both board games and video games), listening to music, hiking, playing basketball, and just relaxing with friends.
Isabella
Pre-Algebra Tutor • +27 Subjects
I am a graduate of MIT. I received my Bachelor of Science in Mathematics with minors in Management Science and Ancient and Medieval Studies. Since graduation, I have started my PhD at Georgia Tech in Operations Research. Throughout my career I have TA'd several math and computer science courses at the college level. I have also taught at summer programs for gifted middle school and high school students. I am passionate about tutoring kids in math and science because I think that a strong foundation in STEM at an early age can set the tone for their future. In my spare time I like to engage in athletics, and was a Division 1 rower in college. Hobbies: reading, swimming, writing, books, music, running, art
Ingrid
Pre-Algebra Tutor • +51 Subjects
I am exploring my creativity by pursuing a double major in Asian Languages and Cultures with a focus in Korean, studying abroad in South Korea as a Benjamin A. Gilman Scholar, leading workshops that teach 3D printing and CAD for undergraduate students as the president of 3D4E, advocating for the first-generation and low-income student community as the Outreach Chair of the Quest+ Scholars Network, and getting involved with the Society of Women Engineers' outreach committee. I currently hold a work-study position as an administrative clerical aide in the Institute of Sustainability and Energy at Northwestern and was an undergraduate researcher in the John Rogers Lab. As I look forward with aspirations of applying to graduate school, areas of research in biomedical engineering and biotechnology that I am particularly interested in include biomaterials, pharmaceuticals, and drug delivery systems. Outside of the classroom, I enjoy learning on my own and sharing my experience and knowledge with my peers and other students. I hope to make use of my experiences with academics and learning in high school and so far in my undergraduate career in order to effectively tutor students who may be experiencing the same struggles in learning that I also experienced.
James
AP Calculus AB Tutor • +40 Subjects
I am currently a senior at Harvard College where I study chemistry, and I'll be attending Columbia Medical School next year. I have years of experience tutoring college students in math (mostly calculus) and chemistry including both general and organic chemistry. In addition, I am very familiar with all sections of the SAT and ACT having prepared several high school students for these tests. I believe that every student is capable of boosting his or her baseline score on these tests, so long as he or she works hard to get to know the format of the tests and the most popular types of questions. I tutor because I love seeing students develop a genuine passion for the subjects they once disliked (such as math and science), once they understand the power of these subjects and their applications to the real world.
Andrew
Pre-Algebra Tutor • +26 Subjects
I am comfortable tutoring math subjects up to multivariable calculus and differential equations, as well as college physics. Hobbies: books, music, art, reading, writing
Asta
Pre-Algebra Tutor • +73 Subjects
I am a graduate of the University of Chicago where I received my undergraduate degree in political science. Right after graduation, I worked as an academic and test prep tutor as well as admissions consultant in Hong Kong. For the past two years, I worked with a number of students to help prepare them for college in the United States.
Sabira
Middle School Math Tutor • +35 Subjects
I am currently attending Johns Hopkins University, pursuing a dual degree in Computer Science and Applied Math and Statistics. I love helping students and I love the feeling I get knowing that I was able to use my knowledge to make someone else happier. My favorite subject to teach is math because there are so many ways to learn it and if one way does not help I can use another. I used to teach taekwondo and interacted with all kinds of students, and I'm excited to help out more! Hobbies: books, reading, music, writing, art
Daniel
Pre-Algebra Tutor • +26 Subjects
I am excited to be home and help fellow straphangers on their educational paths! My largest wealth of tutoring experience is in foreign languages--particularly French--but I also feel very comfortable editing essays of any kind and working through standardized test concepts. My availability is extremely flexible, and anywhere in New York City works for me. I look forward to working with you.
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Frequently Asked Questions
Students often find the transition from Euclidean to affine geometry challenging because affine geometry removes the concept of distance and angle while preserving parallelism and ratios of distances along lines. Common pain points include understanding affine transformations (translations, rotations, and shears) without relying on metric properties, working with affine coordinates and barycentric coordinates, and visualizing how parallel lines remain parallel under affine maps. Many students also struggle with proofs that depend on affine properties rather than congruence or similarity, since the intuition built from classical geometry doesn't directly transfer.
Tutors help students recognize that affine geometry is about structure and relationships that survive transformation, not about measuring distances. A strong approach involves starting with concrete examples—like how a photograph (a perspective projection) preserves collinearity and ratios along lines, even though distances and angles change—then building toward formal definitions of affine spaces and affine maps. Tutors also help students see that many theorems they've proven in Euclidean geometry (like Menelaus' theorem or properties of medians) are actually affine results, which builds confidence that affine geometry extends rather than replaces their prior knowledge.
Affine and barycentric coordinates feel abstract because they don't rely on a fixed origin or orthogonal axes like Cartesian coordinates do. Students must learn that an affine coordinate system is defined by a set of points (an affine frame) rather than a single origin, and that barycentric coordinates express a point as a weighted combination of reference points—a fundamentally different way of locating position. Tutors help by connecting this to familiar ideas: barycentric coordinates are like describing a location as a blend of nearby landmarks, and affine coordinates show how to express any point using a chosen set of reference points and directions.
Affine transformations (which include translations, rotations, scaling, shears, and their compositions) preserve collinearity and parallel lines, but students often memorize this without understanding why. Tutors use visual and kinesthetic approaches—sketching how a shear transformation stretches a square into a parallelogram while keeping opposite sides parallel, or showing how a composition of transformations can be represented as a single matrix equation. By working through examples where students predict what happens to specific points and lines under a transformation, then verify their predictions, they develop the intuition that affine maps are exactly those that preserve the affine structure.
Affine proofs often require students to think about what properties are available—parallelism, collinearity, and ratios—rather than falling back on distance or angle arguments. Tutors teach students to identify the key affine invariants in a problem, then work backward from the conclusion to see which transformations or coordinate systems might simplify the argument. For example, if a problem involves parallel lines and ratios, barycentric coordinates or an affine transformation that simplifies the configuration can make the proof much clearer. Breaking down proofs into "what structure do I need to preserve?" and "which affine tools preserve that?" helps students move past the feeling that affine proofs are mysterious.
Students sometimes conflate affine and projective geometry because both generalize Euclidean geometry, but they do so in different ways: affine geometry preserves parallelism, while projective geometry does not (parallel lines meet at infinity). Tutors clarify this by showing concrete examples—an affine transformation keeps parallel lines parallel, while a perspective projection (projective transformation) does not. Understanding this distinction helps students recognize which tools apply in which context: affine coordinates and affine maps for affine problems, and homogeneous coordinates and projective maps for projective problems. This clarity prevents students from mixing approaches and getting confused about what's being preserved.
Affine geometry appears in computer graphics (texture mapping, sprite transformations), robotics (describing how rigid motions preserve distances but affine transformations model perspective), and engineering (strain and deformation in materials). Tutors can motivate abstract concepts by showing how affine transformations are used in image processing—a shear or rotation applied to an image is an affine map—or how barycentric coordinates are used in 3D graphics to determine whether a point lies inside a triangle. Connecting the theory to applications helps students see affine geometry as a practical toolkit, not just an abstract mathematical structure.
Students benefit from strong linear algebra fundamentals—comfortable with vectors, matrices, and linear transformations—since affine geometry builds on these concepts by adding a translation component. Solid understanding of Euclidean geometry (especially properties of parallel lines, similar figures, and coordinate systems) is also important, because affine geometry is often presented as a generalization that removes metric properties while preserving affine ones. Tutors assess whether gaps in linear algebra or geometry are holding a student back and address them directly, since trying to learn affine geometry without these foundations leads to frustration and surface-level memorization rather than deep understanding.
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