The line of AG(2,9) joining two points
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In collections
Affine planes
This example demonstrates how to construct and display the line joining two given points in the finite affine geometry AG(2,9), i.e., the two-dimensional affine plane over the finite field of order 9. The core mathematical concepts involve finite field arithmetic, affine geometry, coordinate representation of points, and solving linear equations over a finite field. The construction uses GeoGebra commands such as IndexOf, Mod, and Defined to handle modular arithmetic, and defines numerous auxiliary objects (e.g., positionA, ycoordA, xcoef, ycoef, const) to compute the line coefficientsholistically. It then generates multiple points (A1 to A12, B1 to B12) on the line to visualize its structure. This example is suitable for university-level courses in discrete mathematics, abstract algebra, or computational geometry, and serves as a teaching tool for illustrating lines and points in finite geometries.
GeoGebra Commands & Parameters
IndexOf(A, E1:E81) → positionAIndexOf(B, E1:E81) → positionBMod(real(F1), 3) → G1Mod(imaginary(F1), 3) → H1Mod(real(F2), 3) → G2Mod(imaginary(F2), 3) → H2Mod(real(F3), 3) → G3Mod(imaginary(F3), 3) → H3Mod(real(F4), 3) → G4Mod(imaginary(F4), 3) → H4Mod(real(F5), 3) → G5Mod(imaginary(F5), 3) → H5Mod(real(F6), 3) → G6Mod(imaginary(F6), 3) → H6Mod(real(F7), 3) → G7Mod(imaginary(F7), 3) → H7Mod(real(F8), 3) → G8Mod(imaginary(F8), 3) → H8Mod(real(F9), 3) → G9Mod(imaginary(F9), 3) → H9Mod(real(F10), 3) → G10Mod(imaginary(F10), 3) → H10Mod(real(F11), 3) → G11Mod(imaginary(F11), 3) → H11Mod(real(F12), 3) → G12Mod(imaginary(F12), 3) → H12Mod(real(F13), 3) → G13Mod(imaginary(F13), 3) → H13Mod(real(F14), 3) → G14Mod(imaginary(F14), 3) → H14Mod(real(F15), 3) → G15Mod(imaginary(F15), 3) → H15Mod(real(F16), 3) → G16Mod(imaginary(F16), 3) → H16Mod(real(F17), 3) → G17Mod(imaginary(F17), 3) → H17Mod(real(F18), 3) → G18Mod(imaginary(F18), 3) → H18Mod(real(F19), 3) → G19Mod(imaginary(F19), 3) → H19Mod(real(F20), 3) → G20Mod(imaginary(F20), 3) → H20Mod(real(F21), 3) → G21Mod(imaginary(F21), 3) → H21Mod(real(F22), 3) → G22Mod(imaginary(F22), 3) → H22Mod(real(F23), 3) → G23Mod(imaginary(F23), 3) → H23Mod(real(F24), 3) → G24Mod(imaginary(F24), 3) → H24Mod(real(F25), 3) → G25Mod(imaginary(F25), 3) → H25Mod(real(F26), 3) → G26Mod(imaginary(F26), 3) → H26Mod(real(F27), 3) → G27Mod(imaginary(F27), 3) → H27Mod(real(F28), 3) → G28Mod(imaginary(F28), 3) → H28Mod(real(F29), 3) → G29Mod(imaginary(F29), 3) → H29Mod(real(F30), 3) → G30Mod(imaginary(F30), 3) → H30Mod(real(F31), 3) → G31Mod(imaginary(F31), 3) → H31Mod(real(F32), 3) → G32Mod(imaginary(F32), 3) → H32Mod(real(F33), 3) → G33Mod(imaginary(F33), 3) → H33Mod(real(F34), 3) → G34Mod(imaginary(F34), 3) → H34Mod(real(F35), 3) → G35Mod(imaginary(F35), 3) → H35Mod(real(F36), 3) → G36Mod(imaginary(F36), 3) → H36Mod(real(F37), 3) → G37Mod(imaginary(F37), 3) → H37Mod(real(F38), 3) → G38Mod(imaginary(F38), 3) → H38Mod(real(F39), 3) → G39Mod(imaginary(F39), 3) → H39Mod(real(F40), 3) → G40Mod(imaginary(F40), 3) → H40Mod(real(F41), 3) → G41Mod(imaginary(F41), 3) → H41Mod(real(F42), 3) → G42Mod(imaginary(F42), 3) → H42Mod(real(F43), 3) → G43Mod(imaginary(F43), 3) → H43Mod(real(F44), 3) → G44Mod(imaginary(F44), 3) → H44Mod(real(F45), 3) → G45Mod(imaginary(F45), 3) → H45Mod(real(F46), 3) → G46Mod(imaginary(F46), 3) → H46Mod(real(F47), 3) → G47Mod(imaginary(F47), 3) → H47Mod(real(F48), 3) → G48Mod(imaginary(F48), 3) → H48Mod(real(F49), 3) → G49Mod(imaginary(F49), 3) → H49Mod(real(F50), 3) → G50Mod(imaginary(F50), 3) → H50Mod(real(F51), 3) → G51Mod(imaginary(F51), 3) → H51Mod(real(F52), 3) → G52Mod(imaginary(F52), 3) → H52Mod(real(F53), 3) → G53Mod(imaginary(F53), 3) → H53Mod(real(F54), 3) → G54Mod(imaginary(F54), 3) → H54Mod(real(F55), 3) → G55Mod(imaginary(F55), 3) → H55Mod(real(F56), 3) → G56Mod(imaginary(F56), 3) → H56Mod(real(F57), 3) → G57Mod(imaginary(F57), 3) → H57Mod(real(F58), 3) → G58Mod(imaginary(F58), 3) → H58Mod(real(F59), 3) → G59Mod(imaginary(F59), 3) → H59Mod(real(F60), 3) → G60Mod(imaginary(F60), 3) → H60Mod(real(F61), 3) → G61Mod(imaginary(F61), 3) → H61Mod(real(F62), 3) → G62Mod(imaginary(F62), 3) → H62Mod(real(F63), 3) → G63Mod(imaginary(F63), 3) → H63Mod(real(F64), 3) → G64Mod(imaginary(F64), 3) → H64Mod(real(F65), 3) → G65Mod(imaginary(F65), 3) → H65Mod(real(F66), 3) → G66Mod(imaginary(F66), 3) → H66Mod(real(F67), 3) → G67Mod(imaginary(F67), 3) → H67Mod(real(F68), 3) → G68Mod(imaginary(F68), 3) → H68Mod(real(F69), 3) → G69Mod(imaginary(F69), 3) → H69Mod(real(F70), 3) → G70Mod(imaginary(F70), 3) → H70Mod(real(F71), 3) → G71Mod(imaginary(F71), 3) → H71Mod(real(F72), 3) → G72Mod(imaginary(F72), 3) → H72Mod(real(F73), 3) → G73Mod(imaginary(F73), 3) → H73Mod(real(F74), 3) → G74Mod(imaginary(F74), 3) → H74Mod(real(F75), 3) → G75Mod(imaginary(F75), 3) → H75Mod(real(F76), 3) → G76Mod(imaginary(F76), 3) → H76Mod(real(F77), 3) → G77Mod(imaginary(F77), 3) → H77Mod(real(F78), 3) → G78Mod(imaginary(F78), 3) → H78Mod(real(F79), 3) → G79Mod(imaginary(F79), 3) → H79Mod(real(F80), 3) → G80Mod(imaginary(F80), 3) → H80Mod(real(F81), 3) → G81Mod(imaginary(F81), 3) → H81Defined(positionA) → AisproperDefined(positionB) → Bisproper
AI Drawing Prompt Reference
Draw a line in the finite affine plane AG(2,9) passing through two given points. The plane has 9 elements, with each point's coordinates taken from the finite field modulo 3. First, locate the two points, then compute their coordinates modulo 3 to obtain field elements. Next, solve a linear equation over the finite field to determine the line's coefficients. Finally, generate a series of discrete points on the line and display them along with the line itself, illustrating the structure of a line in finite geometry.
The prompt above was compiled from the original drawing commands and is for reference only.