Theoretical arc of goalkeeper position in team sports. Case study: football

The current position of the goalkeeper in football is a function of the course of action and each momentary position of the ball. The goalkeeper monitors the action and the place of a potential shot; however, this shot on goal is a random event. That is why it is so important to take a position in the closest goal area that takes account of the random nature of the ball’s direction. The basic principle of the strategy is to create equal opportunities for defence on the left and on the right. Using the language of geometry, this position can be placed on the bisector of the angle at which the goal can be seen from the potential shooting position. At the beginning of this research paper, the advantages associated with the goalkeeper slightly coming off the goal line along this bisector have been presented, as well as certain restrictions related. Further in the article, the task of determining the theoretical curve along which the goalkeeper should move has been undertaken. Geometrically correct, but unfavourable in practical terms, two circles, Cassini oval, a composition of the arcs of two circles and a segment of the straight line as well as the arc of the ellipse have been considered. In the second part of the paper, the point analysis of the goalkeeper’s position has been changed into real conditions – a defence zone equal to the goalkeeper’s arms’ reach has been taken into account. For these conditions, a curve has been determined, composed of two arcs of the circles and the arc of the ellipse. A detailed analysis has led to the conclusion that the discrepancy between such an arc and a homogeneous arc of the ellipse is practically negligible. Therefore, both arcs: the one composed of parts and the homogeneous one of the ellipse – can be accepted as rational and practically alternative. In this research paper, the problem has been analysed from the geometric point of view, taking into consideration a shot on goal that is a random event for the goalkeeper. The proposed theoretical goalkeeper arc ensures an optimal position, taking account of the shots coming from different directions, from the zone close to the penalty area.