Cutting angle (geometry)

Intersection angle between two straight lines

In geometry, an intersection angle is an angle formed by two intersecting curves or surfaces . When two straight lines intersect, there are generally four intersection angles, two of which are opposite each other and are congruent . The smaller of these two congruent angles is usually referred to as the intersection angle, which is then acute or right-angled . Since secondary angles add up to 180 °, the larger cutting angle, which is then obtuse or right-angled, can be determined from this.

The angles of intersection between the graphs of two real functions can be calculated using the derivatives of the functions at the point of intersection. The angle of intersection between two curves can be determined using the scalar product of the tangential vectors at the point of intersection. The angle of intersection between a curve and a surface is the angle between the tangent vector of the curve and the normal vector of the surface at the point of intersection. The intersection angle of two surfaces is the angle between the normal vectors of the surfaces and then depends on the point on the intersection curve.

Intersection angle of function graphs

Intersection angle between the graphs of two linear functions

The angle of intersection between the graphs of two linear functions with the slopes or is calculated using ${\displaystyle \alpha }$ ${\displaystyle m_{1}}$ ${\displaystyle m_{2}}$

${\displaystyle \tan \alpha =\left|{\frac {m_{1}-m_{2}}{1+m_{1}m_{2}}}\right|}$.

This formula is derived from the addition theorems of the trigonometric functions. If this applies to the slopes , then the tangent function becomes infinite and the two straight lines intersect at right angles . ${\displaystyle m_{1}m_{2}=-1}$

More generally, the intersection angle between the graphs of two differentiable functions with the derivatives or at the point of intersection can also be determined in this way . ${\displaystyle m_{1}}$ ${\displaystyle m_{2}}$

Examples

The graphs of the two linear functions and intersect at this point at an angle, because ${\displaystyle f(x)={\sqrt {3}}x+4}$ ${\displaystyle g(x)={\tfrac {1}{\sqrt {3}}}x-1}$ ${\displaystyle x=-{\tfrac {5}{2}}{\sqrt {3}}}$ ${\displaystyle 30^{\circ }}$

${\displaystyle \tan \alpha =\left|{\frac {{\sqrt {3}}-{\tfrac {1}{\sqrt {3}}}}{1+{\sqrt {3}}\cdot {\tfrac {1}{\sqrt {3}}}}}\right|={\frac {1}{\sqrt {3}}}\Rightarrow \alpha =\arctan {\frac {1}{\sqrt {3}}}=30^{\circ }}$.

The exponential function intersects the constant function at the point at an angle of 45 °, because ${\displaystyle f(x)=e^{x}}$ ${\displaystyle g(x)=1}$ ${\displaystyle x=0}$

${\displaystyle \tan \alpha =\left|{\frac {f'(0)-g'(0)}{1+f'(0)\cdot g'(0)}}\right|=\left|{\frac {1-0}{1+1\cdot 0}}\right|=1\Rightarrow \alpha =\arctan 1=45^{\circ }}$.

Intersection angles of curves and surfaces

Intersection angle of two curves

The angle of intersection of two (here circular) curves is the angle between the tangents of the curves and at the point of intersection . ${\displaystyle A}$ ${\displaystyle B}$ ${\displaystyle P}$

In Euclidean space one can find the intersection angle of two intersecting straight lines with the direction vectors and through ${\displaystyle \alpha }$ ${\displaystyle {\vec {x}}}$ ${\displaystyle {\vec {y}}}$

${\displaystyle \cos \alpha ={\frac {|{\vec {x}}\cdot {\vec {y}}|}{|{\vec {x}}|\,|{\vec {y}}|}}}$

Calculate, where is the scalar product of the two vectors and the Euclidean norm of a vector. More generally, the intersection angle of two differentiable curves can also be determined using the scalar product of the associated tangential vectors and at the intersection point. ${\displaystyle {\vec {x}}\cdot {\vec {y}}}$ ${\displaystyle |{\vec {x}}|}$ ${\displaystyle {\vec {x}}}$ ${\displaystyle {\vec {y}}}$

Examples

The angle of intersection between two intersecting straight lines with the direction vectors and is ${\displaystyle {\vec {x}}=(1,4,5)^{T}}$ ${\displaystyle {\vec {y}}=(2,1,3)^{T}}$

${\displaystyle \cos \alpha ={\frac {|1\cdot 2+4\cdot 1+5\cdot 3|}{{\sqrt {1^{2}+4^{2}+5^{2}}}\,{\sqrt {2^{2}+1^{2}+3^{2}}}}}={\frac {21}{{\sqrt {42}}\,{\sqrt {14}}}}={\frac {3}{2{\sqrt {3}}}}\Rightarrow \alpha =\arccos {\frac {3}{2{\sqrt {3}}}}=30^{\circ }}$.

In order to calculate the intersection angle between the straight line and the unit circle at the point , the two tangential vectors at this point are determined as and and thus ${\displaystyle {\sqrt {3}}x-y=1}$ ${\displaystyle x^{2}+y^{2}=1}$ ${\displaystyle ({\tfrac {\sqrt {3}}{2}},{\tfrac {1}{2}})}$ ${\displaystyle {\vec {x}}=(1,{\sqrt {3}})}$ ${\displaystyle {\vec {y}}=(-{\tfrac {1}{2}},{\tfrac {\sqrt {3}}{2}})}$

${\displaystyle \cos \alpha ={\frac {|1\cdot (-{\tfrac {1}{2}})+{\sqrt {3}}\cdot {\tfrac {\sqrt {3}}{2}}|}{{\sqrt {1+3}}\,{\sqrt {{\tfrac {1}{4}}+{\tfrac {3}{4}}}}}}={\frac {|-{\tfrac {1}{2}}+{\tfrac {3}{2}}|}{2\cdot 1}}={\frac {1}{2}}\Rightarrow \alpha =\arccos {\frac {1}{2}}=60^{\circ }}$.

Intersection angle of a curve with a surface

Intersection angle , line g, plane E, projection line p ${\displaystyle \alpha }$
${\displaystyle \gamma =\beta =90^{\circ }-\alpha \Rightarrow \sin(\alpha )=\cos(90^{\circ }-\gamma )={\frac {|n\cdot x|}{|n||x|}}}$

The angle of intersection between a straight line with the direction vector and a plane with the normal vector is through ${\displaystyle \alpha }$ ${\displaystyle {\vec {x}}}$ ${\displaystyle {\vec {n}}}$

${\displaystyle \sin \alpha ={\frac {|{\vec {n}}\cdot {\vec {x}}|}{|{\vec {n}}|\,|{\vec {x}}|}}}$

given. More generally, it is also possible to calculate the angle of intersection between a differentiable curve and a differentiable surface using the scalar product of the tangential vector of the curve with the normal vector of the surface at the point of intersection. This angle of intersection is then equal to the angle between the tangential vector of the curve and its orthogonal projection onto the tangential plane of the surface. ${\displaystyle {\vec {x}}}$ ${\displaystyle {\vec {n}}}$

Intersection angle of two surfaces

Cutting angle between two planes: ${\displaystyle \alpha =\beta =\gamma }$

The angle of intersection between two planes with the normal vectors and is corresponding ${\displaystyle \alpha }$ ${\displaystyle {\vec {n}}}$ ${\displaystyle {\vec {m}}}$

${\displaystyle \cos \alpha ={\frac {|{\vec {n}}\cdot {\vec {m}}|}{|{\vec {n}}|\,|{\vec {m}}|}}}$.

More generally, the angle of intersection between two differentiable surfaces can also be determined in this way. This angle of intersection generally depends on the point on the intersection curve.

See also

• Dangerous place
• Intersection line

literature

• Rolf Baumann: Geometry: Trigonometric functions, trigonometry, addition theorems, vector calculation . Mentor 1999, ISBN 3580636367 , pp. 76-77
• Andreas Filler: Elementary Linear Algebra . Springer, 2011, ISBN 9783827424136 , pp. 159-161
• Cutting angle In: Schülerduden - Mathematics II . Bibliographisches Institut & FA Brockhaus, 2004, ISBN 3-411-04275-3 , pp. 361–362

Web links

Commons : cut angles  - collection of images, videos and audio files

• Eric W. Weisstein : Line-Line Angle . In: MathWorld (English).
• J. Pahikkala, Chi Woo: Angle between two lines . In: PlanetMath . (English)