Busy. Please wait.

show password
Forgot Password?

Don't have an account?  Sign up 

Username is available taken
show password


Make sure to remember your password. If you forget it there is no way for StudyStack to send you a reset link. You would need to create a new account.
We do not share your email address with others. It is only used to allow you to reset your password. For details read our Privacy Policy and Terms of Service.

Already a StudyStack user? Log In

Reset Password
Enter the associated with your account, and we'll email you a link to reset your password.
Don't know
remaining cards
To flip the current card, click it or press the Spacebar key.  To move the current card to one of the three colored boxes, click on the box.  You may also press the UP ARROW key to move the card to the "Know" box, the DOWN ARROW key to move the card to the "Don't know" box, or the RIGHT ARROW key to move the card to the Remaining box.  You may also click on the card displayed in any of the three boxes to bring that card back to the center.

Pass complete!

"Know" box contains:
Time elapsed:
restart all cards
Embed Code - If you would like this activity on your web page, copy the script below and paste it into your web page.

  Normal Size     Small Size show me how


Distance for spherical geometry d(A, B) := ∠AOB · R
Polar triangle A'=Pol(BC) and ∠AOA' ≤ π/2
Bipolar theorem If A'B'C'=Pol(ABC), then ABC=Pol(A'B'C') Angles are π − a, π − b, π − c and sides are π − α, π − β, π − γ
Thales theorem The base angles of a spherical isosceles triangle are equal
Area of a spherical triangle (α + β + γ − π)R^2
Sum of angles of a spherical triangle π < α + β + γ ≤ 3π
Sum of sides of a spherical triangle 0 < a + b + c ≤ 2π
Isometries of spherical geometry Rotation, reflection and glide reflection
Homothety A map which is a scalar multiplication
Pappus' theorem Let a and b be lines with points A1,A2,A3 and B1,B2,B3. Pi=AjBknAkBj. Then the points Pi, Pj and Pk are collinear.
Pascal's theorem If A, B, C, D, E, F lie on a conic then the points AB ∩ DE, BC ∩ EF, CD ∩ F A are collinear.
Desargues' theorem If the lines joining the vertices of triangles A1A2A3 and B1B2B3 intersect at one point S, then the intersection points Pi = AjAk ∩ BjBk are collinear.
Distance for Klein model d(A, B) = 1/2|ln|[A, B, X, Y ]||
Parabolic Mobius transformation Conjugate to z → z+1
Elliptic Mobius transformation Conjugate to z → az, |a| = 1, two similar fixpoints
Hyperbolic Mobius transformation Conjugate to z → az, |a| =/ 1 and a ∈ R, two different fixpoints
Loxodromic Mobius transformation Conjugate to z → az, two different fixpoints
Inversion Takes A to A' lying on the ray OA s.t. |OA|·|OA'|=r^2
Distance for Poincare disc model d(A, B)=|ln|[A, B, X, Y ]||
Isometries of Poincare disc model Mobius transformations, inversions, reflections preserving the disc
Every isometry of the Poincare disc model can be written as az+b/cz+d or az~+b/cz~+d
Sum of angles in a hyperbolic triangle Less than pi
Distance for upper half-plane model d(A, B)=|ln|[A, B, X, Y ]||
Every isometry of the upper half-plane model can be written as az+b/cz+d or a(-z~)+b/c(-z~)+d
Area of hyperbolic triangle π−(α+β+γ)
Area of n-gon (n−2)π−sum(αi)
Distance for two-sheet hyperboloid model d(A, B) = 1/2|ln|[A, B, X, Y ]||
Reflection in hyperboloid model ra : x → x−2((x,a)/(a,a))a
Elliptic isometry 1 fixpoint in H^2
Parabolic isometry 1 fixpoint in ∂H^2
Hyperbolic isometry 2 fixpoints in ∂H2
Created by: Marge-Homer