Group Abstract

Message Boards

WOLFRAM COMMUNITY

763 Views

8 Replies

3 Total Likes

View groups...

Follow this post

Share this post:

GROUPS:

Mathematics Geometry Wolfram Language

Solve geometric quantities using GeometricScene

Xiangyang Zhou

Posted 2 months ago

I constructed the following GeometricScene with parameters {s,r}. The scene consists of a circle of radius r, and an equilateral triangle {a,b,c} inscribed in the circle. s being the length of one side of the triangle. I want to calculate the value of s, dependent on the value of r. scene=GeometricScene[{{a,b,c,o},{s,r}},{CircleThrough[{a,b,c},o,r],GeometricAssertion[Triangle[{a,b,c}],"Equilateral"],s==TriangleMeasurement[{a,b,c},"Perimeter"]}]; RandomInstance[scene] s/.%["Quantities"] It returns 4.41219 as the value of s. I'm guessing behind the scenes, Mathematica chose a value for r and use that value for s. Is there any way to get an analytical solution of s dependent on r?

POSTED BY: Xiangyang Zhou

8 Replies

Sort By:

Gianluca Gorni

Gianluca Gorni, University of Udine

Posted 2 months ago

`Eliminate` gives necessary conditions, not always sufficient. Try this: GeometricScene[{{a, b, c, o}, {s, r}}, {CircleThrough[{a, b, c}, o, r], GeometricAssertion[Triangle[{a, b, c}], "Equilateral"], s == TriangleMeasurement[{a, b, c}, "Perimeter"], s != 3 Sqrt[3] r}]["Conclusions"]

Eliminate gives necessary conditions, not always sufficient. Try this:

GeometricScene[{{a, b, c, o}, {s, r}}, {CircleThrough[{a, b, c}, o, r],
    GeometricAssertion[Triangle[{a, b, c}], "Equilateral"], 
   s == TriangleMeasurement[{a, b, c}, "Perimeter"], 
   s != 3 Sqrt[3] r}]["Conclusions"]

POSTED BY: Gianluca Gorni

Xiangyang Zhou

Posted 2 months ago

POSTED BY: Xiangyang Zhou

Xiangyang Zhou

Posted 2 months ago

POSTED BY: Xiangyang Zhou

Ian Ford

Ian Ford, Wolfram Research

Posted 2 months ago

`"Quantities"` gave the values for the particular instance of the scene found by `RandomInstance`. You can use `GeometricSolveValues` on the abstract scene instead: scene = GeometricScene[{{a, b, c, o}, {s, r}}, {CircleThrough[{a, b, c}, o, r], GeometricAssertion[Triangle[{a, b, c}], "Equilateral"], s == TriangleMeasurement[{a, b, c}, "Perimeter"]}]; GeometricSolveValues[scene, s] {ConditionalExpression[s, Sqrt[(Indexed[a, {1}] - Indexed[b, {1}])^2 + (Indexed[a, {2}] - Indexed[b, {2}])^2] + Sqrt[(Indexed[a, {1}] - Indexed[c, {1}])^2 + (Indexed[a, {2}] - Indexed[c, {2}])^2] + Sqrt[(Indexed[b, {1}] - Indexed[c, {1}])^2 + (Indexed[b, {2}] - Indexed[c, {2}])^2] == s && (Indexed[a, {1}] - Indexed[b, {1}])^2 + (Indexed[a, {2}] - Indexed[b, {2}])^2 == (Indexed[b, {1}] - Indexed[c, {1}])^2 + (Indexed[b, {2}] - Indexed[c, {2}])^2 == (Indexed[a, {1}] - Indexed[c, {1}])^2 + (Indexed[a, {2}] - Indexed[c, {2}])^2 && (Indexed[a, {1}] - Indexed[o, {1}])^2 + (Indexed[a, {2}] - Indexed[o, {2}])^2 == (Indexed[b, {1}] - Indexed[o, {1}])^2 + (Indexed[b, {2}] - Indexed[o, {2}])^2 == (Indexed[c, {1}] - Indexed[o, {1}])^2 + (Indexed[c, {2}] - Indexed[o, {2}])^2 == r^2 && r > 0]}

"Quantities" gave the values for the particular instance of the scene found by RandomInstance. You can use GeometricSolveValues on the abstract scene instead:

scene = GeometricScene[{{a, b, c, o}, {s, 
     r}}, {CircleThrough[{a, b, c}, o, r], 
    GeometricAssertion[Triangle[{a, b, c}], "Equilateral"], 
    s == TriangleMeasurement[{a, b, c}, "Perimeter"]}];

GeometricSolveValues[scene, s]

{ConditionalExpression[s, 
  Sqrt[(Indexed[a, {1}] - Indexed[b, {1}])^2 + (Indexed[a, {2}] - 
        Indexed[b, {2}])^2] + 
     Sqrt[(Indexed[a, {1}] - Indexed[c, {1}])^2 + (Indexed[a, {2}] - 
        Indexed[c, {2}])^2] + 
     Sqrt[(Indexed[b, {1}] - Indexed[c, {1}])^2 + (Indexed[b, {2}] - 
        Indexed[c, {2}])^2] == 
    s && (Indexed[a, {1}] - Indexed[b, {1}])^2 + (Indexed[a, {2}] - 
       Indexed[b, {2}])^2 == (Indexed[b, {1}] - 
       Indexed[c, {1}])^2 + (Indexed[b, {2}] - 
       Indexed[c, {2}])^2 == (Indexed[a, {1}] - 
       Indexed[c, {1}])^2 + (Indexed[a, {2}] - 
       Indexed[c, {2}])^2 && (Indexed[a, {1}] - 
       Indexed[o, {1}])^2 + (Indexed[a, {2}] - 
       Indexed[o, {2}])^2 == (Indexed[b, {1}] - 
       Indexed[o, {1}])^2 + (Indexed[b, {2}] - 
       Indexed[o, {2}])^2 == (Indexed[c, {1}] - 
       Indexed[o, {1}])^2 + (Indexed[c, {2}] - Indexed[o, {2}])^2 == 
    r^2 && r > 0]}

GeometricSolveValues output for abstract scene