Documentation

class Point

Attributes

• x: The x-coordinate of the point.
• y: The y-coordinate of the point.
• cordinates: A tuple representing the x and y coordinates.
• quadrant: The quadrant of the Cartesian plane the point lies in.

Methods

• __init__(x, y):

  Initializes the Point object with x and y coordinates.
  

• __str__() -> str:

  Returns a string representation of the coordinates.
  

• isorigin() -> bool:

  Checks if the point is the origin.
  
  Returns:
      bool: True if the point is the origin, False otherwise.
  

• distancePoint(point: 'Point') -> sp.Expr:

  Calculate the Euclidean distance to another point.
  
  Args:
      point (Point): The other point to calculate the distance from.
  
  Returns:
      sp.Expr: The distance as a SymPy expression.
  

• distanceLine(line: 'Line') -> sp.Expr:

  Calculate the perpendicular distance from the point to a line.
  
  Args:
      line (Line): The line to calculate the distance from.
  
  Returns:
      sp.Expr: The distance as a SymPy expression.
  

• ison(element) -> bool:

  Determine if the point lies on a given geometric element.
  
  Args:
      element (Point | Line | Segment): The geometric element.
  
  Returns:
      bool: True if the point lies on the element.
  

• findPoint(line, point, distance) -> tuple[Point, Point | None]

  Finds two points on a given line at a specific distance from a reference point.
  
  Args:
      line (Line): The line on which to find the points.
      point (Point): The reference point.
      distance (int | float | str | sp.Expr): The desired distance from the reference point.
  
  Returns:
      tuple[Point, Point | None]: A tuple containing two points at the given distance.
  
  Raises:
      ValueError: If there are no possible points at the given distance.
  

Example

from mathworld import Point, Line

# Create a Point object at coordinates (3, 4)
point1 = Point(3, 4)
print(point1)  # Expected output: (3, 4)

# Check if point1 is the origin
is_origin = point1.isorigin()
print(is_origin)  # Expected output: False

# Create another Point object at (0, 0)
point2 = Point(0, 0)

# Calculate the distance between point1 and point2
distance = point1.distancePoint(point2)
print(distance)  # Expected output: 5

# Test findPoint with a line and a point
point3 = Point(3, 2)
line = Line('y = -x + 5')

# Find one of the two points on the line at distance of 2 from pointA
points_at_distance = Point.findPoint(line, point3, 2)
print(points_at_distance[0])  # Expected output: (3 - sqrt(2), sqrt(2) + 2)

class Line

Attributes

• equation: The equation of the line (can be explicit or implicit).
• implicitEquation: The implicit equation of the line.
• slope: The slope of the line.
• intercept: The y-intercept of the line.
• a, b, c: Coefficients for the implicit line equation ax + by + c = 0.

Methods

• __init__(equation):

  Initializes the Line object from an equation.
  
  Args:
      equation (str | sp.Equality): The equation defining the line.
  
  Raises:
      ValueError: If the provided equation format is invalid.
  

• __str__() -> str:

  Returns the equation of the line as a string.
  

• isHorizontal() -> bool:

  Check if the line is horizontal.
  
  Returns:
      bool: True if the line is horizontal.
  

• isVertical() -> bool:

  Check if the line is vertical.
  
  Returns:
      bool: True if the line is vertical.
  

• isParallel(line: 'Line') -> bool:

  Check if the line is parallel to another line.
  
  Args:
      line (Line): The other line.
  
  Returns:
      bool: True if the lines are parallel.
  

• isPerpendicular(line: 'Line') -> bool:

  Check if the line is perpendicular to another line.
  
  Args:
      line (Line): The other line.
  
  Returns:
      bool: True if the lines are perpendicular.
  

• intersection(line: 'Line') -> Point:

  Calculate the intersection point with another line.
  
  Args:
      line (Line): The other line.
  
  Returns:
      Point: The intersection point.
  

• isPerpendicularBisector(segment: 'Segment') -> bool:

  Check if the line is the axis of a given segment.
  
  Args:
      segment (Segment): The segment to check.
  
  Returns:
      bool: True if the line is the axis of the segment.
  

• isBisector(line1: 'Line', line2: 'Line') -> bool:

  Check if the line is the bisector of two lines.
  
  Args:
      line1 (Line): The first line.
      line2 (Line): The second line.
  
  Returns:
      bool: True if the line is the bisector of the two lines.
  

• findParallel(point: Point) -> 'Line':

  Find a parallel line that passes through a given point.
  
  Args:
      point (Point): The point through which the parallel line passes.
  
  Returns:
      Line: The parallel line.
  

• findPerpendicular(point: Point) -> 'Line':

  Find a perpendicular line that passes through a given point.
  
  Args:
      point (Point): The point through which the perpendicular line passes.
  
  Returns:
      Line: The perpendicular line.
  

• findBisector(line: 'Line') -> tuple['Line', 'Line']:

  Find the bisectors of the angles formed between the current line and another line.
  
  Args:
      line (Line): The other line.
  
  Returns:
      tuple[Line, Line]: The two angle bisectors as lines.
  

• findLine(point1, point2, slope, intercept) -> Line

  Generate a line based on given parameters.
  
  Args:
      point1 (Point | None): The first point on the line.
      point2 (Point | None): The second point on the line.
      slope (int | float | str | sp.Expr | None): The slope of the line.
      intercept (int | float | str | sp.Expr | None): The y-intercept of the line.
  
  Returns:
      Line: The constructed line object.
  
  Raises:
      ValueError: If sufficient parameters are not provided.
  

Example

from mathworld import Point, Line

# Create a Line object from an equation
line_eq = Line('y - 1 = 2x')
print(line_eq)  # Expected output: y = 2*x + 1

# Check if line_eq is horizontal
is_horizontal = line_eq.isHorizontal()
print(is_horizontal)  # Expected output: False

# Create another Line object representing y = -x + 5
line2 = Line('y = -x + 5')

# Find intersection of two lines
intersection_point = line_eq.intersection(line2)
print(intersection_point)  # Expected output: (4/3, 11/3)

# Create two Point objects to define a line
pointA = Point(1, 1)   # First point at (1, 1)
pointB = Point(4, 3)   # Second point at (4, 3)

# Using findLine with two points to create a line equation
line_eq = Line.findLine(pointA, pointB)
print(line_eq)  # Expected output: y = 2*x/3 + 1/3

class Segment

Attributes

• point1: The first endpoint of the segment.
• point2: The second endpoint of the segment.
• length: The length of the segment.
• middle: The midpoint of the segment.
• line: The line containing the segment.
• perpendicularBisector: The perpendicular bisector of the segment.

Methods

• __init__(point1: Point, point2: Point):

  Initializes the Segment object with two points.
  
  Args:
      point1 (Point): The first endpoint of the segment.
      point2 (Point): The second endpoint of the segment.
  

Example

from mathworld import Point, Segment

# Create two Point objects for segment endpoints
pointA = Point(1, 2)
pointB = Point(4, 6)

# Initialize a Segment object with these points
segment = Segment(pointA, pointB)
print(segment.length)  # Expected output: 5

Constants

• ORIGIN: A predefined point at (0, 0).
• X_AXIS: Line representing the x-axis.
• Y_AXIS: Line representing the y-axis.
• BISECTOR_1_3, BISECTOR_2_4: Lines representing specific angle bisectors of quadrants.