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Example Relations

🐌: Takes a long time to terminate.

Basic Examples

sin(x) = cos(y)

y - x = sin(exp(x + y))

x^2 + y^2 = 1 ∨ y = -cos(x)

Algebraic Equations

(2y-x-1)(2y-x+1)(2x+y-1)(2x+y+1)((5x-2)^2+(5y-6)^2-10)((5x)^2+(5y)^2-10)((5x+2)^2+(5y+6)^2-10)=0

((x-2)^2+(y-2)^2-0.4)((x-2)^2+(y-1)^2-0.4)((x-2)^2+y^2-0.4)((x-2)^2+(y+1)^2-0.4)((x-2)^2+(y+2)^2-0.4) ((x-1)^2+(y-2)^2-0.4)((x-1)^2+(y-1)^2-0.4)((x-1)^2+y^2-0.4)((x-1)^2+(y+1)^2-0.4)((x-1)^2+(y+2)^2-0.4) (x^2+(y-2)^2-0.4)(x^2+(y-1)^2-0.4)(x^2+y^2-0.4)(x^2+(y+1)^2-0.4)(x^2+(y+2)^2-0.4) ((x+1)^2+(y-2)^2-0.4)((x+1)^2+(y-1)^2-0.4)((x+1)^2+y^2-0.4)((x+1)^2+(y+1)^2-0.4)((x+1)^2+(y+2)^2-0.4) ((x+2)^2+(y-2)^2-0.4)((x+2)^2+(y-1)^2-0.4)((x+2)^2+y^2-0.4)((x+2)^2+(y+1)^2-0.4)((x+2)^2+(y+2)^2-0.4) = 0

Examples taken from GrafEq

  • 📂 Single Relation/Abstract/Simple/

    • 📄 Irrationally Contin.gqs 🐌🐌🐌

      y = gcd(x, 1)

    • 📄 Parabolic Waves.gqs

      |sin(sqrt(x^2 + y^2))| = |cos(x)|

    • 📄 Prime Bars.gqs

      gcd(⌊x⌋, Γ(⌊sqrt(2⌊x⌋) + 1/2⌋)) ≤ 1 < x - 1

    • 📄 Pythagorean Pairs.gqs

      ⌊x⌋^2 + ⌊y⌋^2 = ⌊sqrt(⌊x⌋^2 + ⌊y⌋^2)⌋^2

    • 📄 Pythagorean Triples.gqs

      ⌊x⌋^2 + ⌊y⌋^2 = 25

    • 📄 Rational Beams.gqs

      gcd(x, y) > 1

  • 📂 Single Relation/Abstract/Traditionally Difficult/

    • 📄 Infinite Frequency.gqs

      y = sin(40 / x)

    • 📄 O Spike.gqs

      (x (x-3) / (x-3.001))^2 + (y (y-3) / (y-3.001))^2 = 81

    • 📄 Solid Disc.gqs

      81 - x^2 - y^2 = |81 - x^2 - y^2|

    • 📄 Spike.gqs

      y = x (x-3) / (x-3.001)

    • 📄 Step.gqs

      y = atan(9^9^9 (x-1))

    • 📄 Upper Triangle.gqs

      x + y = |x + y|

    • 📄 Wave.gqs

      y = sin(x) / x

  • 📂 Single Relation/Enumerations/Binary/

    • 📄 binary naturals.gqs

      (1 + 99 ⌊mod(⌊y⌋ 2^⌈x⌉, 2)⌋) (mod(x,1) - 1/2)^2 + (mod(y,1) - 1/2)^2 = 0.15 ∧ ⌊-log(2,y)⌋ < x < 0

    • 📄 binary squares.gqs

      (1 + 99 ⌊mod(⌊y⌋^2 2^⌈x⌉, 2)⌋) (mod(x,1) - 1/2)^2 + (mod(y,1) - 1/2)^2 = 0.15 ∧ x < 0 < ⌊y⌋^2 ≥ 2^-⌈x⌉

  • 📂 Single Relation/Enumerations/Decimal/

    • 📄 decimal squares.gqs

      mod(if(30 max(|mod(y,1) - 1/2|, |mod(x,0.8)+0.1 - 1/2| + |mod(y,1) - 1/2| - 1/4) < 1, 892, if(30 max(|mod(y,1) - 1/10|, |mod(x,0.8)+0.1 - 1/2| + |mod(y,1) - 1/10| - 1/4) < 1, 365, if(30 max(|mod(y,1) - 9/10|, |mod(x,0.8)+0.1 - 1/2| + |mod(y,1) - 9/10| - 1/4) < 1, 941, if(30 max(|mod(x,0.8) + 0.1 - 4/5|, |mod(y,1) - 7/10| + |mod(x,0.8) + 0.1 - 4/5| - 1/8) < 1, 927, if(30 max(|mod(x,0.8) + 0.1 - 1/5|, |mod(y,1) - 7/10| + |mod(x,0.8) + 0.1 - 1/5| - 1/8) < 1, 881, if(30 max(|mod(x,0.8) + 0.1 - 1/5|, |mod(y,1) - 3/10| + |mod(x,0.8) + 0.1 - 1/5| - 1/8) < 1, 325, if(30 max(|mod(x,0.8) + 0.1 - 4/5|, |mod(y,1) - 3/10| + |mod(x,0.8) + 0.1 - 4/5| - 1/8) < 1, 1019, sqrt(-1)))))))) 2^-⌊mod(⌊y⌋^2 / 10^-⌈1.25x⌉, 10)⌋, 2) ≥ 1 ∧ x < 0 < ⌊y⌋^2 ≥ 10^-⌈1.25x⌉

  • 📂 Single Relation/Enumerations/Trees/

    • 📄 bi-infinite binary tree.gqs

      sin(2^⌊y⌋ x + π/4 (y - ⌊y⌋) - π/2) = 0 ∨ sin(2^⌊y⌋ x - π/4 (y - ⌊y⌋) - π/2) = 0

  • 📂 Single Relation/Half-Toned/

    • 📄 Simply Spherical.gqs

      sin(20x) - cos(20y) + 2 > 4 if((x+1)^2 + (y-1)^2 < 25, (3/4 - 1/15 sqrt((x+4)^2 + (y-3)^2)), (0.65 + 1/π atan(6 (sqrt((x-1)^2/30 + (y+1)^2/9) - 1))))

    • 📄 Tube.gqs

      cos(5x) + cos(5/2 (x - sqrt(3) y)) + cos(5/2 (x + sqrt(3) y)) > 1 + if((x^2 + 2y^2 - 1600) (x^2 + 3 (y-2)^2 - 700) ≤ 0, 3/2 sin(1/4 sqrt((x+3)^2 + 2 (y-3)^2)), 2 atan(1/8 sqrt(4 (x-2)^2 + 10 (y+4)^2) - 9)^2)

  • 📂 Single Relation/Linelike/

    • 📄 Frontispiece #2.gqs

      x / cos(x) + y / cos(y) = x y / cos(x y) ∨ x / cos(x) + y / cos(y) = -(x y / cos(x y)) ∨ x / cos(x) - y / cos(y) = x y / cos(x y) ∨ x / cos(x) - y / cos(y) = -(x y / cos(x y))

    • 📄 Frontispiece.gqs

      x / sin(x) + y / sin(y) = x y / sin(x y) ∨ x / sin(x) + y / sin(y) = -(x y / sin(x y)) ∨ x / sin(x) - y / sin(y) = x y / sin(x y) ∨ x / sin(x) - y / sin(y) = -(x y / sin(x y))

    • 📄 Hair.gqs 🐌

      sin((x + sin(y)) (sin(x) + y)) = cos(sin((sin(x) + cos(y)) (sin(y) + cos(x)))) ∨ sin((x + sin(y)) (sin(x) + y)) = cos(sin((sin(x) + cos(y)) (sin(y) - cos(x)))) ∨ sin((x + sin(y)) (sin(x) + y)) = cos(sin((sin(x) - cos(y)) (sin(y) + cos(x)))) ∨ sin((x + sin(y)) (sin(x) + y)) = cos(sin((sin(x) - cos(y)) (sin(y) - cos(x)))) ∨ sin((x + sin(y)) (sin(x) - y)) = cos(sin((sin(x) + cos(y)) (sin(y) + cos(x)))) ∨ sin((x + sin(y)) (sin(x) - y)) = cos(sin((sin(x) + cos(y)) (sin(y) - cos(x)))) ∨ sin((x + sin(y)) (sin(x) - y)) = cos(sin((sin(x) - cos(y)) (sin(y) + cos(x)))) ∨ sin((x + sin(y)) (sin(x) - y)) = cos(sin((sin(x) - cos(y)) (sin(y) - cos(x)))) ∨ sin((x - sin(y)) (sin(x) + y)) = cos(sin((sin(x) + cos(y)) (sin(y) + cos(x)))) ∨ sin((x - sin(y)) (sin(x) + y)) = cos(sin((sin(x) + cos(y)) (sin(y) - cos(x)))) ∨ sin((x - sin(y)) (sin(x) + y)) = cos(sin((sin(x) - cos(y)) (sin(y) + cos(x)))) ∨ sin((x - sin(y)) (sin(x) + y)) = cos(sin((sin(x) - cos(y)) (sin(y) - cos(x)))) ∨ sin((x - sin(y)) (sin(x) - y)) = cos(sin((sin(x) + cos(y)) (sin(y) + cos(x)))) ∨ sin((x - sin(y)) (sin(x) - y)) = cos(sin((sin(x) + cos(y)) (sin(y) - cos(x)))) ∨ sin((x - sin(y)) (sin(x) - y)) = cos(sin((sin(x) - cos(y)) (sin(y) + cos(x)))) ∨ sin((x - sin(y)) (sin(x) - y)) = cos(sin((sin(x) - cos(y)) (sin(y) - cos(x))))

    • 📄 Highwire.gqs

      |x cos(x) - y sin(y)| = |x cos(y) - y sin(x)|

    • 📄 Trapezoidal Fortress.gqs

      |x cos(x) + y sin(y)| = x cos(y) - y sin(x)

  • 📂 Single Relation/Solid/

    • 📄 Sharp Threesome.gqs

      sin(sqrt((x+5)^2 + y^2)) cos(8 atan(y / (x+5))) sin(sqrt((x-5)^2 + (y-5)^2)) cos(8 atan((y-5) / (x-5))) sin(sqrt(x^2 + (y+5)^2)) cos(8 atan((y+5) / x)) > 0

    • 📄 The Disco Hall.gqs

      sin(|x + y|) > max(cos(x^2), sin(y^2))

Examples taken from [Tup01]

  • Fig. 15 (b) “the patterned star”

    0.15 > |rankedMin([cos(8y), cos(4(y-sqrt(3)x)), cos(4(y+sqrt(3)x))], 2) - cos(⌊3/π mod(atan2(y,x),2π) - 0.5⌋) - 0.1| ∧ rankedMin([|2x|, |x-sqrt(3)y|, |x+sqrt(3)y|], 2) < 10

Examples taken from GrafEq Reviews

y = sqrt(x)^2

y = sqrt(x-1) / sqrt(x-3)

The graph must be empty:

y = sqrt(x-3) sqrt(1-x)

Examples taken from Cool Graphs of Implicit Equations

🐌

exp(sin(x) + cos(y)) = sin(exp(x + y))

sin(sin(x) + cos(y)) = cos(sin(x y) + cos(x))

sin(x^2 + y^2) = cos(x y)

|sin(x^2 - y^2)| = sin(x + y) + cos(x y)

|sin(x^2 + 2 x y)| = sin(x - 2 y)

Tests for Conjunction and Disjunction

The graph must be empty:

y = x ∧ y = x + 0.00001

The graph must not be empty:

y = x ∨ y = x + 0.00001

y < sqrt(x) ∧ y < sqrt(-x)

y < sqrt(x) ∨ y < sqrt(-x)

y = sin(40 / x) ∧ (x > 0 ∧ y > 0)

y = sin(40 / x) ∧ (x > 0 ∨ y > 0)

y = sin(40 / x) ∨ (x > 0 ∧ y > 0)

y = sin(40 / x) ∨ (x > 0 ∨ y > 0)