Biarc Construction Analise de indeterminações e condições de não solução

Indetermination in biarc construction Asked 1 month ago Modified 29 days ago 30 times Viewed 0 Im following friendly article in order to compute a biarc in The author writes this R2 Choosing a value for plays a big role in the shape of the biarc Negative values will make it long and curved Values near zero will make the first arc compact d1 Some values can even create a case without a solution The main ideia we construct two circunferences that are tangent to each other and from five inputs a starting point and its tangent an ending point and its tangent and a scalar please see the diagram C1 C2 P1 t1 P2 t2 d1 The author wrote the tangent point as a linear combination of and before finding pm C1 C2 d1 d2 d2 pm P1 d1t1 d2 d1 d2 P2 d2t2 d1 d1 d2 He also wrote a nicer version of pm pm p1 c1t1 q d1t1 t2 t2 Where q P2 P1 There are a couple of denominator indeterminations that are confusing me The first one is for the scalar d2 d2 2 q q d1 q t1 2 2 1 q t2 d1 t2 t1 Anoter two are the centers of both circunferences and C1 C2 C1 p1 pm p1 pm p1 2 pm p1 n1 n1 C2 p2 pm p2 pm p2 2 pm p2 n2 n2 Where is the vector with at Also the term multipliying is the radius ni ni ti 0 Pi ni Ci From the input values and Id like to know which input value settings cause indetermination in the denominators along the construction of the biarc the cases without a solution P1 P2 t1 t2 d1 ie Ron pointed out in the comments that I need to better define when a biarc is invalid I dont know exactly but I think its fine to have zero radius because it could produce straight segments connecting the points and The possibility of an infinite radius would be my second question because I think it might lead to a line that overshoots the points when the circunferences curvature changes sign P1 P2 κ 1r After following the article I wrote the procedure described in it to produce animation just press play this Thank you for your time geometry vectors analyticgeometry Share Cite Edit Delete Flag edited May 31 at 024 Ethan Bolker 875k 6 100 183 asked May 26 at 1615 ivansnpmaster 1 2 I think that you will have to better define when a solution is invalid because mathematicians tend to be more forgiving than the general public when thinking about such things For example if one of the arcs has a zero radius is that OK If both circles have infinite radii and are therefore straight lines and only wrap around via the point at infinity is that OK Ron Kaminsky May 26 at 1959 You should also edit the question to actually include and not just link to the definition of the biarc interpolation which you are talking about External links tend to sometimes rot and SE prefers that that wont cause questions to become impossible to understand Ron Kaminsky May 26 at 2032 Thank you for the response RonKaminsky I modified my question with more context ivansnpmaster May 31 at 003 Indeterminação na construção do bianc Sejam P1 um ponto inicial e t1 sua tangente P2 um ponto final e t2 sua tangente Seja pmC1C2 o ponto de interseção entre ar circunferencias Sejam q1 e q2 os pontos de interseção das retas que passam por P1 e P2 nas direções de t1 e t2 com a reta tangente de C1 e C2 Sejam também d1q1P1 d2q2P2 Logo q1p1 d1t1 q2p2 d2 t2 1 Por outro lado consideremos as coordenadas de q1 q2 pm Então cosθ p1mq12d2 q11p1md1 d1 p1m d1 q12 d2 q11 d2 p1m d1 d2 p1m d2 q11 d1 q12 2 e senθ p2m q22d2 q21 p2md1 d1 p2m d1 q22 d2 q21 d2 p2m d1 d2 p2m d2 q21 d1 q22 3