Lets say you have marked a point A on the center line and want to use it to find the tangent line from there to a circle centered around point C. These points on the circle were NOT the point where the tangent line hits a circle, but only a step used to find the required point on the center line. Make SURE when you use your point on the line through the centers of the circle that you have erased the perpendicular lines AND the points on the circle circumferences where the perpendicular intersected the circle. The line continued on through the point on the center line to the opposite circle will also be a tangent.Īgain the problem now boils down to drawing a tangent line to a circle given a point external to the circle that the tangent must pass through. Use this point to draw the tangent to either circle. Mark clearly this point where the line crosses the center line. This line will cross the line through the centers. This time pick a radius along this perpendicular line where one radius is on each side of the line through the centers.ĭetermine the points where the radii you have drawn hits their respective circles.ĭraw a line from each of these two points. These same lines will also be tangent to the far circle.įor an internal bisector, draw the perpendicular to the line through the center of each circle Use the vanishing point to draw the tangents to the nearby circle and continue that line on to the far circle. The point you just determined is a Vanishing Point.Įrase all of the extra lines and points except for the circles, line through the centers and vanishing point. The new line through these two points will hit the line through the center of the circles beyond the smaller circle. TO draw external tangents pick a radius from each circle that is perpendicular to the line through the centers and have both radii on the same side of the center line.ĭraw a line and determine the equation of the line using where the radii hit the circle. Note the points where the vertical lines intersect the circle. Given two circles, determine the internal and external tangents.ĭetermine a line through the center of the circles.Īt the center of both circles determine a perpendicular line $$ =n\implies nx-y 12-11n=0$$ where $n$ is the gradientĬan you follow the method used in Case $1$ to find the two values of $n,$ which implies there will be two simple common tangents A circle with equation $(x-x_c)^2 (y-y_c)^2 - r^2 = 0$ is represented by a 3x3 matrix as You can approach this with homogeneous coordinates.
0 Comments
Leave a Reply. |