geometry question solver
Python
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README.md

geosolver

NOTE: SOME SECTIONS OF THIS README FILE ARE OUT OF DATE! I WILL UPDATE IT AS SOON AS POSSIBLE. THANK YOU!

geosolver is an end-to-end system that solves high school geometry questions. That is, its input is question text in natural language and diagram in raster graphics, and its output is the answer to the question.

geosolver is divided into four core parts: diagram praser, text parser, joint parser, and solver. Text parser transforms the text in natural language into a logical form. Diagram parser extracts information from the diagram. Joint parser combines the results of the text parser and the diagram parser and outputs the final logical form. Solver accepts the logical form from the joint parser and outputs the answer. This tutorial will first walkthrough each part independently (the modules except the joint parser can be used independently), and in the last section it will show how to connect them for an end-to-end system.

Accessing questions via GeoServer

Location: geosolver.database

Required 3rd-party packages: requests

Every question is a tuple of key, text, words, diagram path, and choices (see geosolver.database.states.Question). You can either define it yourself, or more easily, you can download it via GeoServer, a web interface for managing and displaying geosolver database. Please refer to the GeoServer repository to see how to set it up.

  1. Make sure that geosolver.settings.GEOSERVER_URL has a correct URL. Note that by default this is pointing to a local server, because I am hosting a local server myself.
  2. Import the geoserver interface (an instance of geosolver.database.geoserver_interface.GeoServerInterface):

    from geosolver import geoserver_interface
  3. To download all questions tagged 'test' and print the text of each, type:

    questions = geoserver_interface.download_questions('test')
    for id_, question in questions.iteritems():
      print(question.text)

    Note that the object returned by geoserver_interface.download_questions is a dict object.

  4. To download a single question with specific id (e.g. 1037), type:

    questions = geoserver_interface.download_questions(1037)
    print(questions.values()[0].text)

    Note that, regardless of the number of questions returned by geoserver_interface.download_questions, the returned object is always a dict object (with possibly single element, like above).

  5. You can access other properties of the question by question.words, question.diagram_path, and question.choices. See Diagram parser section to learn how to use question.diagram_path, and see Text parser section to learn how to use question.words and question.choices.

Diagram parser

Location: geosolver.diagram

Required 3rd-party packages: numpy, OpenCV 3.0.0 or higher (cv2)

Diagram parsing consists of five finer steps: image segment parsing, primitive paring, primitive selecting, core parsing, and graph parsing. We will explain each of them in detail below. You can also refer to geosolver.diagram.run_diagram module to see full examples corresponding to these.

Parsing image segments

Image segment parsing is the task of obtaining the diagram segment (and label segments) from the original image. For instance, given an original image

original image

we obtain the diagram segment

diagram segment

To do so, obtain a Question object (question) as shown above and run the following:

from geosolver.diagram.parse_image_segments import parse_image_segments

image_segment_parse = parse_image_segments(open_image(question.diagram_path))
image_segment_parse.diagram_image_segment.display_binarized_segmented_image()
for idx, label_image_segment in image_segment_parse.label_image_segments.iteritems():
  label_image_segment.display_segmented_image()

This is equivalent to geosolver.diagram.run_diagram.test_parse_image_segments.

Parsing primitives from the diagram segment

Primitive parsing is the task of obtaining over-gernerated, noisy primitives from the diagram segment. For instance, given the diagram segment above, we want to obtain

primitive parse

To do so:

from geosolver.diagram.parse_primitives import parse_primitives

primitive_parse = parse_primitives(image_segment_parse)
primitive_parse.display_primitives()

This is equivalent to geosolver.diagram.run_diagram.test_parse_primitives.

Selecting primitives

We want to select a subset of the over-generated primitives that accurately represents the diagram:

selected primitives

To do so:

from geosolver.diagram.select_primitives import select_primitives

selected = select_primitives(primitive_parse)
selected.display_primitives()

This is equivalent to geosolver.diagram.run_diagram.test_select_primitives.

Parsing core

CoreParse consists of the critical (intersection) points and the circles in the diagram. It can be thought as the post-processed parse of the selected primitive parse. By parsing the core, we obtain:

core

in which the blue dots are the intersection points.

To do so:

from geosolver.diagram.parse_core import parse_core

core_parse = parse_core(selected)
core_parse.display_points()

This is equivalent to geosolver.diagram.run_diagram.test_parse_core.

Parsing graph

GraphParse contains high level information such as whether a line exists between two points, etc. To obtain the graph parse, run:

from geosolver.diagram.parse_graph import parse_graph
from geosolver.diagram.get_instances import get_all_instances

graph_parse = parse_graph(diagram_parse)
lines = get_all_instances(graph_parse, 'line')
circles = get_all_instances(graph_parse, 'circle')
arcs = get_all_instances(graph_parse, 'arc')
angles = get_all_instances(graph_parse, 'angle')
print("Displaying lines...")
for key, line in lines.iteritems():
  graph_parse.display_instances([line])
print("Displaying circles...")
for key, circle in circles.iteritems():
  graph_parse.display_instances([circle])
print("Displaying arcs...")
for key, arc in arcs.iteritems():
  graph_parse.display_instances([arc])
print("Displaying angles...")
for key, angle in angles.iteritems():
  graph_parse.display_instances([angle])

This is equivalent to geosolver.diagram.run_diagram.test_parse_graph.

Text parser

Solver

Location: geosolver.solver

Required 3rd-party packages: numpy, scipy

For all the examples, see geosolver.solver.run_solver. We will go through its first example here.

Example 0

"In triangle ABC, AB = x, BC = 3, CA = 4. Which of the following is a possible value for x? (A) 5 (B) 8"

To solve this question, we first need to define geosolver.solver.variable_handler.VariableHandler:

from geosolver.solver.variable_handler import VariableHandler
vh = VariableHandler()

The variable handler allows us to define variables in the (first sentence):

x = vh.number('x')
A = vh.point('A')
B = vh.point('B')
C = vh.point('C')
AB = vh.line(A, B)
BC = vh.line(B, C)
CA = vh.line(C, A)

Use vh.number to define a numeric variable (with name as argument), vh.point to define a point (with name as argument), and vh.line to define a line (with two end points as arguments).

Next, we define the relations (information) mentioned in the question (AB = x, BC = 3, CA = 4):

c = vh.apply('LengthOf', AB)
a = vh.apply('LengthOf', BC)
b = vh.apply('LengthOf', CA)
p1 = vh.apply('Equals', a, 3)
p2 = vh.apply('Equals', b, 4)
p3 = vh.apply('Equals', c, x)

p1, p2, and p3 are the three prior relations in the text.

geosolver.solver.numeric_solver.NumericSolver allows us to reconstruct the geometry according to the given information:

from geosolver.solver.numeric_solver import NumericSolver
ns = NumericSolver(vh, [p1, p2, p3])

NumericSolver can analyze whether an input relation is simultaneously satisfiable with the prior relations via find_assignment method. It returns a satisfying assignment if they are simultaneously satisfiable, and it returns None if not. That is,

q1 = vh.apply('Equals', x, 5)
print(ns.find_assignment(q1))

will print out an assignment, and

q2 = vh.apply('Equals', x, 8)
print(ns.find_assignment(q2))

will print out None.

Note that you can use q1 = x == 5 instead of vh.apply('Equals', x, 5). Other operators such as <, >, +, , -, /, ** (power) are supported as well. Geometric relations, such as Perpendicular, Tangent, etc., require vh.apply to be used. For a complete list of usable relations, refer to geosolver.ontology.ontology_semantics (To be updated soon!).