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Browse College Euclidean Geometry

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Angles as Turns [05/29/2003]
How can angles be negative?

Angles of Reflection [03/13/2001]
I am trying to prove that the incoming angle of a ray from one focus within a ellipse is equal to the angle of the outgoing ray to the other focus...

Area, Circumference of an Ellipse [7/29/1996]
How do I calculate the area and circumference of a given ellipse?

Are Angles Dimensionless? [08/31/2003]
If you look at the dimensions in the equation arc length = r*theta, it appears that angles must be dimensionless. But this can't be right. Or can it?

Area of a Crescent [06/18/2001]
When observing a total eclipse of the sun we need to determine the area of the sun that has not been covered by the moon.

Area of an Ellipse Cut by a Chord [05/26/2000]
How can you calculate area of the part of an ellipse cut off by a chord, if you know the major and minor axes, and the chord?

Area of an Ellipsoid [09/28/2001]
How do you calculate (or even estimate) the area of an ellipsoid that is neither oblate nor prolate?

Area of a Reuleaux Triangle [06/13/2002]
Could you please help me find a formula to find the area of a Reuleaux triangle?

Ball Bouncing off a Line Segment [05/04/2001]
If you take an arbitrary line on a 2D plane, e.g. x1y1 - x2y2, then take a point that moves about the plane, say pxpy, can you tell if this point has crossed the line at any time?

Barycentric Calculus [01/06/1999]
How does barycentric calculus compare with trilinear or cartesian calculus?

Best-fitting Line to a Number of Points [04/25/2001]
I have a number of points on a plane and want to find a line that best- fits through the points, minimizing the sum of squares of the distances of each point from the line.

Bretschneider's Theorem and Cyclic Quadrilaterals [11/30/2000]
Can you prove Bretschneider's Theorem for the area of a quadrilateral? Also, can you show that any quadrilateral with supplementary opposing angles can be inscribed in a circle?

Catenary Curve [03/30/1999]
Find the vertex of a catenary curve.

Centroid - Center of Gravity [03/25/2002]
Can a triangle have a unique centre of gravity?

Circle Slicing Cylinder: 3 Dimensions, 3 Approaches [08/10/2016]
A C++ programmer seeks ways to find the points where a cylinder and circle might intersect in 3-space. With parameterizations aplenty, Doctor George suggests three possible methods.

Classifying Shape Based on Coordinate Points [01/03/2000]
How can I design an algorithm to classify shapes based on a relatively small set of (x,y) coordinates that describe the boundary of a closed object?

Connected Sets in Topology [04/22/1998]
Exploring connected sets with examples in Euclidean space.

Constructing the Trisection of an Angle [07/20/2008]
Why can you construct the trisection of some angles and not of others?

Coordinate Geometry - Goat in a Circular Field [8/22/1996]
A goat is tethered by a rope of length L to a point on the circumference of a circular field of radius R. Find L in terms of R if the goat can graze exactly half the area of the field.

Curve Fitting Algorithm [6/20/1996]
Can you explain "least squares approximation"?

Definitions of Edge and Face in 2D and 3D [10/10/2008]
What is the 'official' definition of 'edge'? Specifically, is an edge restricted to the intersection of two non-coplanar faces or do two- dimensional shapes have edges? I'm also curious about a definition of 'face'. How many faces does a two-dimensional shape have?

Deriving a 2D Rotation Matrix [11/17/2009]
How can I geometrically derive this rotation matrix?
  |x'| = |cos(theta)   -sin(theta)| |x|
  |y'|   |sin(theta)    cos(theta)| |y|

Deriving Parametric Equations for Cissoid of Diocles [02/12/2005]
How do you derive the parametric equations for the Cissoid of Diocles?

Desargues' Theorem [07/03/1998]
Why is Desargues' two triangle theorem easy to prove in three dimensions but impossible in two dimensions?

Determining Length of Material Remaining on a Roll [11/24/2003]
Is there a mathematical formula to determine the length of material on a roll, given the outside diameter of the core, the outside diameter of the whole roll, and the thickness of the material (determined by a micrometer)?

Distance between Points of Tangency [02/16/2003]
Two circles, one of radius 5, the other of radius 8, intersect at exactly one point, and the center of each circle lies outside the other circle. A line is externally tangent to both circles. Find the distance between the two points of tangency.

Drawing an Ellipse Using Conjugate Diameters [12/07/2005]
Can you explain how to use Conjugate Diameters to draw an ellipse?

Euclid's Fifth Postulate [6/24/1996]
I am interested in finding some theorems, axioms, or postulates similar to Euclid's Fifth Postulate.

Finding Intersections of Two Ellipses [06/17/2005]
As part of a computer program, I'm working on the problem of detecting the intersection of two ellipses given the length of their axes and coordinates of their centers on the X-Y plane. The ellipses can be oriented so that their major axis is either vertical or horizontal. Is there a way to do this that does not require computationally expensive techniques such as brute force checking of points?

Finding Reflection Points within a Rectangle [03/24/2005]
Given a beginning point and an ending point inside a rectangle, I'm trying find a formula, algorithum or calculation that would tell me where on the rectangle I would have to aim a laser from the given beginning point so that it reflects off exactly two sides of the rectangle and connects with the given ending point.

Finding the Center of a Circle [12/26/1996]
Given a circle of radius R with center point unknown, a line with equation Y = mx+b and a line at Y = -.08, find the x,y coordinates of the points of tangency where the two lines intersect the circle.

Find the Perimeter of the Rectangle [09/06/2002]
Two circles of radii 9 and 17 centimetres are enclosed within a rectangle with one side of length 50 cm. The two circles touch each other, and each touches two adjacent sides of the rectangle. Find the perimeter of the rectangle.

Fitting an Arc to a Point and Two Tangent Lines [11/14/2004]
I'm wondering how to fit an arc to a point and two circles. There is a vertical line segment of length 0.3 m, with a circle of radius 0.037 m coming from the midpoint of that line, and another circle of radius 0.104 m coming from the bottom of the segment. I want to fit a 3 point arc to the top of the segment and the tangents of the two circles.

Folium of Descartes and Parametric Equations [11/23/1998]
How do you plot an implicit function, such as the folium of Descartes, with the equation y^3 + x^3 = 3xy?

Formula for Radial Latitudes and Longitudes [10/15/2002]
I want to make a radius search tool that returns all places based on latitude/longitude within a certain radius of a given place.

General Equation for Intersections of Line and Ellipse [03/13/2007]
I am trying to find a general equation for the intersection points (if there are any) of a line and ellipse. The line is defined by two points (x0,y0) and (x1,y1) and the ellipse is defined as x^2/a^2 + y^2/b^2 = r^2. I tried writing an equation of the line and then substituting it into the ellipse equation, but it got too complicated.

The Goat In the Field Problem [05/24/1997]
A farmer tethers a goat to the circumference of a circular field. What ratio of field radius to length of rope must he use so that the goat can graze only half the area?

Goat Tied by a 10-Meter Rope [11/28/2001]
A goat is tied to the corner of a 5-by-4-meter shed by a 10-meter rope. What area is grazed by the goat? If the shed is a circle with radius r, and the rope is 2r, what is the area grazed?

Golden Spiral [03/23/1998]
What is the equation of the Golden Spiral?

Hyperbolic Geometry [03/24/2003]
Explain this assumption: Assuming that Euclidean geometry is consistent, had any of the failed attempts to prove Euclid's 5th Postulate from the other axioms succeeded, they would have actually completely destroyed Euclidean geometry as a consistent body of thought.

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