February 2009 Archives

Try dragging the points.

The dot product is a powerful vector operation and one of its most handy usage is to compute an angle between two lines.

The dot product in 2D

A·B = a1 * b1 + a2 * b2 = ||A|| * ||B|| * cos(θ) 

• A and B are 2D vectors and A = [a1 a2], B = [b1, b2]
• ||A|| represents is the the magnitude of vector A
• θ is the angle between A and B

In the sample movie, the line AB represents the vector A and CD represents the vector B

sample movie and scripts

Try dragging the points A and B. The line perpendicularly intersects(CD) is depicted using normal vector of AB

The normal vector in 2d is obtained by exchanging x and y-components and multiplying -1 to the y-component.

sample movie and scripts

This sample demonstrates two ways of ratating coordinate around a center point

The first example(P1 and C1) is straight forward while the second(P2 and C2) is more versatile and can be more efficent when treating more than one point in the same angle, in that case sine and cosine need to be computed only once before applied to each coordinate calculation of the points.

// P1 and C1 
var angle:Number = Math.atan2(P1.y - C1.y,  P1.x - C1.x);
var radius:Number = Point.distance(C1, P1);
P1.x = C1.x + Math.cos(angle + vr) * radius;
P1.y = C2.y + Math.sin(angle + vr) * radius;

// P2 and C2 
var distX:Number = P2.x - C2.x;
var distY:Number = P2.y - C2.y;
P2.x =  Math.cos(vr) * distX - Math.sin(vr) * distY + C2.x;
P2.y =  Math.cos(vr) * distY + Math.sin(vr) * distX + C2.y;

In action script 3, the Matrix class's rotate method works in the same way as the second example.

sample movie and scripts

The angles of a triangle(OAB) can be computed if the length of three sides are known using transformed version of cosine formula. Note that the angle between line L and OB is also needed to obtain actual angles of the segments.

Low of cosines

OAB = 
Math.acos((OA * OA + AB * AB - BO * BO) / 
(2 * OA * AB));

BOA = 
Math.acos((OA * OA + BO * BO - AB * AB) / 
(2 * OA * BO));

ABO = 
Math.acos((AB * AB + BO * BO - OA * OA) / 
(2 * BO* AB));

See 'Law of cosines - Wikipedia' for more about cosine formura.

sample movie and scripts

This sample demonstrates how trigonometry and pythagorean theorem work, which are essencial to deal with distances and angles in 2D or 3D space. Try dragging the point P.

Pythagorean theorem

OP = Math.sqrt(handle.x * handle.x + handle.y * handle.y);

Trigonometry

sin(AOC) = AC;
cos(AOC) = OC;
sin(AOC) = AC / OC;

note that the direction of the Y-coordinate in Flash is opposite of Cartesian coordinates.

See 'Trigonometry - Wikipedia' for more about trigonometry and 'Pythagorean theorem - Wikipedia' for Pythagorean theorem.

sample movie and scripts

The law of cosines is a statement about a general triangle which relates the lengths of its sides to the cosine of one of its angles.

The formula can be used to compute the angles of a triangle if the three sides are known. Try dragging points A, B and C.

Law of cosines

Math.pow(CA, 2) = Math.pow(AB, 2) + Math.pow(BC, 2) - 2 * AB * BC * cos(ABC);

ABC = Math.acos((AB * AB + BC * BC - CA * CA) / (2 * AB * BC)); 
BCA = Math.acos((BC * BC + CA * CA - AB * AB) / (2 * BC * CA)); 
CAB = Math.acos((AB * AB + CA * CA - BC * BC) / (2 * AB * CA)); 

See 'Law of cosines - Wikipedia' for more about law of cosines.

sample movie and scripts

This is a simple model to demonstrate how to simulate gravity.

In this example, gravity is not mutual but the circle A is fixed in a place and the circle B flies around it under the effect of gravity. The product of the first three terms(G * m1 * m2) in the Newton's law is simplified and set to 10, which is enough for rough simulation like this.
Try dragging A to change the orbit of B.

Newton's law of universal gravitation (in AS3)

F = G * m1 * m2 / Math.pow(r, 2);

• F is the magnitude of the gravitational force between the two point masses
• G is the gravitational constant (6.674 × 10⁻¹¹ N m² kg^-2)
• m1 is the mass of the first point mass
• m2 is the mass of the second point mass
• r is the distance between the two point masses

See 'Newton's law of universal gravitation - Wikipedia' for more about law of universal gravitation.

sample movie and scripts