In this paper, some results involving isoptic curves and constant \(\phi \)-width curves are given for any closed curve. The non-convex case, as well as non-simple shapes with or without cusps are considered. Relating the construction of isoptics to the construction given in Holditch’s theorem, a kind of curves is defined: the isochordal-viewed curves. The explicit expression of these curves is given together with some examples. Integral formulae on the area of their isoptics are obtained and a Barbier-type theorem is derived. Finally, a characterization for isochordal-viewed hedgehogs and curves of constant \(\phi \)-width is given in terms of an angle function.

在本文中，对于任何闭合曲线，给出了一些涉及等光曲线和恒定\(\phi \)-宽度曲线的结果。考虑非凸面情况，以及带或不带尖头的非简单形状。将等光体的构造与霍尔迪奇定理中给出的构造联系起来，定义了一种曲线：等弦观察曲线。这些曲线的显式表达与一些例子一起给出。获得了其等光面面积的积分公式，并导出了巴比尔型定理。最后，根据角度函数给出了等弦观察刺猬和恒定\(\phi \)宽度曲线的特征。